QA 

50 1 


276   174 


Manual  of 
Descriptive  Geometrj 


Wald 


o 


'r- 


CQ 


IN  MEMORIAM 
FLOR1AN  CAJORI 


A  MANUAL 


DESCRIPTIVE  GEOMETRY, 


NUMEROUS  PROBLEMS. 


BY 


CLARENCE  A.  WALDO,  A.M., 

i/ 

PROFESSOR  OF  MATHEMATICS,  PURDUE  UNIVERSITY, 
LAFAYETTE,  IND. 


BOSTON: 

PUBLISHED  BY  D.   C.   HEATH  &  CO. 
1895. 


COPTBIGHT,   1887, 

BY  CLARENCE  A.  WALDO. 


J.  8.  CUSHING  &  Co.,  PRINTERS,  BOSTON. 


PREFACE. 


T^vESCRIPTIVE  GEOMETRY  gives  power  to  express  con- 
ceptions and  to  solve  problems  in  the  constructive  arts ; 
it  also  effectively  disciplines  the  geometrical  imagination. 

To  accomplish  these  ends,  nothing  is  better  than  problems  of 
progressive  difficulty,  which,  taken  in  their  logical  order,  the 
student  can  master  alone,  or  with  the  aid  of  a  small  amount  of 
judicious  suggestion,  and  this  principle  has  controlled  the  plan 
of  this  book.  Part  I.,  therefore,  consists  exclusively  of  prob- 
lems systematically  arranged.  The  Introduction  should  be 
read  before  the  student  undertakes  to  solve  these,  as  it  is 
not  intended  for  recitation,  but  for  a  preparatory  lecture  and  for 
reference.  It  is  not  expected  that  any  student  will  solve  all  the 
problems,  nor  would  it  be  a  wise  expenditure  of  time.  A 
course  has  been  laid  down,  —  by  no  means  a  minimum  one, — 
and  in  Part  II.  of  the  book,  suggestions,  analyses,  and  occa- 
sional demonstrations  for  the  solution  of  the  problems  of  this 
course  have  been  given,  with  the  intention,  however,  of  always 
leaving  some  real  work  for  the  student.  A  large  number  of 
additional  problems  have  been  stated,  which  can  be  substituted 
at  will  for  the  others  or  can  be  used  independently.  Such 
combinations  of  the  problems  can  be  readily  formed  that  the 
instructor  may  have  from  year  to  year  the  substantial  advan- 
tages of  a  change  of  text-book. 


IV  PREFACE. 

Part  III.  is  a  condensed  statement  designed  for  occasional 
reference  in  the  earlier  part  of  the  work,  but  especially  as  a 
review  before  leaving  the  subject  or  in  preparing  for  examination. 

In  using  the  material  provided  in  this  book,  the  author  has 
found  a  method  somewhat  as  follows  productive  of  the  best 
results :  Out  of  every  three  exercises,  one,  an  hour  in  length, 
is  spent  with  the  class  in  explaining  difficulties,  in  opening  new 
phases  of  the  subject,  and  in  pointing  out  short  and  elegant 
methods  of  solution,  based  as  far  as  possible  upon  the  discov- 
eries of  the  class.  As  often  as  seems  necessary,  analyses  and 
geometrical  reasons  are  called  for.  The  other  two  exercises, 
each  two  hours  in  length,  are  spent  by  the  students  in  work 
under  the  eye  of  the  instructor,  in  solving  and  reporting  prob- 
lems, and  receiving  such  assistance  as  seems  necessary  or  judi- 
cious. 

When  the  constructions  have  been  approved,  the  student 
copies  and  arranges  them,  and  prepares  a  suitable  index  and 
title-page.  The  set  of  solutions  thus  formed  is  then  perma- 
nently bound,  and  in  the  end  becomes  the  property  of  the 
student  who  makes  it. 

The  following  are  the  special  features  of  the  book : 

First.  The  method  of  unfolding  the  subject  by  problems 
systematically  arranged,  and  supplemented  by  suggestions 
when  needed. 

Second.    The  large  number  of  problems  given. 

Third.  The  method  of  stating  the  problems,  which  in  con- 
nection with  the  notation  adopted  makes  every  lettered  drawing 
entirely  self-explanatory. 

Fourth.  The  introduction  of  several  subjects  of  consider- 
able descriptive  value,  such  as  the  axis  of  affinity,  axonometry, 
v  Pascal's  and  Brianchon's  hexagrams. 


PREFACE.  V 

Fifth.  The  early  discussion  of  the  cone  and  cylinder  of  rev- 
olution, and  the  sphere,  in  order  that  from  the  beginning  these 
surfaces  may  be  used  as  auxiliary. 

Sixth.  The  omission  of  all  plates  except  a  few  of  a  generic 
character. 

It  has  been  the  intention  of  the  author  to  prepare  a  book 
that  will  stimulate  the  student  and  can  safely  be  left  in  his 
hands  at  all  times,  in  the  same  way  that  a  book  of  directions 
may  be  left  in  the  hands  of  a  student  in  a  zoological  labora- 
tory. It  is  hoped  that  any  one  of  three  classes  of  teachers  of 
the  subject  will  find  it  serviceable  : 

First.  Those  who  believe  it  necessary  to  continue  the  meth- 
ods of  demonstration  peculiar  to  Ancient  Geometry  through 
the  course  in  Descriptive  Geometry,  but  wish  to  supplement 
this  work  with  practical  exercises. 

Second.  Those  who  prefer  the  lecture  system  rather  than  the 
use  of  text-books,  but  desire  a  book  of  exercises  for  the  syste- 
matic grounding  of  their  students  in  the  elements. 

Third.  Those  who  try  to  find  in  these  pages  all  they  need 
for  a  short,  thorough  course  in  the  fundamental  principles  of 
Descriptive  Geometry. 

The  book  is  intended  for  the  class-room,  but  it  is  believed 
that  the  industrious  student  will  be  able  to  master  it  by  him- 
self. 

Several  books  in  German  have  been  freely  drawn  upon  for 
problems,  though  many  of  them  were  collected  while  the 
author  was  attending  a  course  of  lectures  upon  the  subject,  by 
Professor  Marx,  of  the  Royal  Polytechnic  School  at  Munich, 
and  some  are  entirely  original.  All,  however,  have  been  re- 
arranged and  recast  to  suit  the  requirements  of  the  present 
work.  Pohlke  has  been  freely  consulted  in  the  preparation  of 


VI  PREFACE. 

Part  III.,  though  the  works  of  De  la  Gournerie,  Mannheim, 
Delabar,  Gugler,  Fiedler,  Steiner  and  others  have  been  at  hand 
for  reference. 

I  wish  to  express  my  thanks  to  Pres.  T.  C.  Mendenhall, 
of  the  Rose  Polytechnic  Institute,  and  to  my  associate  in  the 
Faculty,  Prof.  W.  L.  Ames,  of  the  department  of  Mechanical 
Drawing,  both  of  whom  read  my  manuscript  and  made  valu- 
able and  helpful  suggestions ;  also  to  Mr.  E.  G.  Waters,  a 
student  of  Rose  Polytechnic,  who  has  aided  me  in  the  prepara- 
tion of  the  plates  for  this  work. 

C.  A.  WALDO. 

TERRE  HAUTE,  IND., 
June  17, 1887, 


TABLE  OF  CONTENTS. 


INTRODUCTION. 

PAGE 

The  purpose  and  place  of  Descriptive  Geometry         ....  1 

Previous  knowledge  assumed 1 

How  to  solve  its  problems 2 

Preliminary  apparatus  and  definitions 2 

The  projection  of  a  point,  line,  surface,  or  solid          ....  3 

Transformation  of  fundamental  planes 4 

Discussion  of  a  preliminary  exercise 4 

Recapitulation 6 

Notation  and  abbreviations  employed 6 


PART  I. 


EXERCISES  AND  PROBLEMS. 


SECTION  I. 
Point,  Line  and  Plane. 

PROBLEMS.  PAGE 

1-2.  Representation  of  a  point         .        .        .        .        .        .11 

3-10.  Construction  of  right  lines 11 

11-12.  Description  of  cone  and  cylinder  of  revolution  and  sphere  12 

13-20.  Right  line 12 

21.  Point  and  right  line 13 

24-30.  Plane .        .  13 

23,  31-36.  Plane  and  point  therein 13 

22,  37.  Plane  and  line  therein 13 

38,  39.  Two  intersecting  lines  in  a  plane 13 

40.  Two  parallel  lines  in  a  plane 14 

41-45.  Point  and  right  line  in  the  plane 14 


Vlll 


TABLE   OF   CONTENTS. 


PROBLEMS. 

46,  47.  Plane  and  point  without  it        ... 

48.  Plane  and  right  line  without  it 

49-53.  Plane  and  point  and  line  without  it  . 

54,  55.  Two  intersecting  planes     .... 

56.  Two  parallel  planes 

57-60.  Plane  and  non-intersecting  lines  in  space 

61-66.  Two  planes,  point  and  line 

67-74.  Trihedral  angle 

75,  77.  Bounded  plane  figure        .... 

76.  Axis  of  affinity 

78-85.  Pyramid 

86,  87.  Regular  polyhedrons          .... 

88-91.  Plane  section  of  polyhedrons     . 

92,  93.  Intersection  of  right  line  and  polyhedrons 

94-96.  Intersections  of  polyhedrons     . 

97-104.  Axonometry 


PAGE 
14 
14 

14 
15 
15 
15 
15 
16 
16 
16 
16 
17 
18 
18 
18 
18 


SECTION  II. 
Additional  exercises  on  the  Point,  Line  and  Plane 


19 


SECTION  III. 
Lines  and  Surfaces  of  an  Order  Higher  than  the  First. 

171-178.  Lines  of  the  second  order 25 

179-182.  Cone  and  cylinder 26 

183-186.  Development  of  cone  and  cylinder,  together  with  plane 

section 26 

187-189.  Piercing  of  cone,  cylinder  and  sphere  .  .  .  .27 

190-201.  Tangent  plane  to  cone,  cylinder  and  developable  surface  27 
202-207.  Intersection  of  solids,  at  least  one  in  each  case  being 

cone  or  cylinder 28 

208-214.  Tangent  plane  to  surfaces  of  revolution  ....  28 

215.  Tangent  cone  to  ellipsoid 29 

216,  217.  Surfaces  of  revolution  intersected  by  solids  ...  29 

218-224.  Warped  surfaces 29 


SECTION  IV. 

Additional  exercises  on  Lines  and  Surfaces  of  an  Order  Higher  than 

the  First  30 


TABLE   OF   CONTENTS.  IX 

PART  II. 

SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS. 
SECTION  I. 

Point,  Line  and  Plane. 

PAGE 

Problems  1-104 36 

Axis  of  affinity  defined  and  its  existence  demonstrated       ...  45 

Axonometry 50 

SECTION  II. 
Lines  and  Surfaces  of  an  Order  Higher  than  the  First. 

Problems  171-223 52 

Conic  section  inscribed  in  a  parallelogram  as  the  locus  of  a  system 

of  points 52 

Conic  section  inscribed  in  a  parallelogram  as  the  envelope  of  a  sys- 
tem of  lines 53 

Pascal's  hexagram .55 

Brianchon's  hexagram 57 

Development  of  surfaces 58 

Demonstration  for  the  hyperboloid  of  revolution  of  one  sheet    .        .  61 

PART    III. 

SUMMARY  OF  PRINCIPLES  AND  DEFINITIONS. 
SECTION  I. 
Projections. 

ARTICLE.  PAGE 

1.  Descriptive  Geometry 65 

2.  Shading  and  perspective 65 

3.  Representation  of  a  solid 65 

4.  Central  projection 65 

5.  Parallel  projection 65 

6.  Oblique  parallel  projection 65 

7.  Orthogonal  parallel  projection 66 

8.  Restricted  meaning  of  Descriptive  Geometry  .         .        .        .66 

9.  Planes  of  projection 66 

10.  Ground  line 66 

11.  Revolutions                                                                                   .  66 


TABLE   OF   CONTENTS. 


SECTION  II. 
Point,  Line  and  Plane. 

ARTICLE.  PAGB 

12.  Projection  of  a  point 66 

13.  Projection  of  a  right  line 67 

14,  15.     Traces  of  a  line 67 

16.  Angle  of  inclination 67 

17.  Projection  of  parallel  right  lines      ......  67 

18.  Plane,  how  represented 67 

19.  Plane,  how  determined .  67 

20.  Point  in  plane 67 

21.  Line  in  plane .  67 

22.  Plane  perpendicular  to  H  or  V 67 

23.  Line  perpendicular  to  plane 67 

24.  Two  parallel  planes 68 


SECTION  III. 
Line  in  General. 

25.  Line,  how  generated 68 

26.  Plane  curve 68 

27.  Space  curve .  68 

28.  Classification  of  plane  curves 68 

29.  Higher  plane  curves 68 

30.  Transcendental  curve 68 

31.  Tangent 69 

32.  Normal 69 

33.  Axis  and  vertex 69 

34.  Diameter 69 

35.  Osculating  circle  and  radius  of  curvature        .        .        .        .69 

36.  Number  of  space  curves 69 

37.  Helix 69 

38.  Spherical  epicycloid 69 

SECTION  IV. 
Surfaces  in  General. 

39.  .  Surface 70 

40.  Directrix,  directer 70 

41.  Kinds  of  surfaces                                                                           ,  70 


TABLE   OF   CONTENTS.  XI 

ARTICLE.  PAGE 

42.  Algebraical  surfaces 70 

43.  Families  of  surfaces 70 

44.  Ruled  surfaces .  70 

45.  Double  curved  surfaces 71 

SECTION  V. 
Developable  Surfaces. 

46.  Developable  surface  further  defined 71 

47.  Developable  surface  with  helical  directrix       .        .        .        .71 

48.  Special  developable  surface 71 

49.  Cone 72 

50.  Cylinder 72 

51.  Tangent  plane 72 

52.  Shortest  path .        .72 

SECTION  VI. 

Surfaces  of  Revolution. 

53.  Surface  of  revolution  defined 72 

54.  Axis 72 

55.  Parallels 73 

56.  Meridian 73 

57.  Equator 73 

58.  Representation  of  surfaces  of  revolution 73 

59.  Circle  of  the  gorge 73 

60.  Orders  of  surfaces  of  revolution 73 

61.  Subdivision  of  surfaces  of  revolution 73 

62.  Tangent  plane 74 

SECTION  VII. 
Warped  Surfaces. 

63.  Number  of  warped  surfaces 74 

64.  Laws  for,  how  expressed 75 

65.  Simplest  law 75 

66.  Property  of  a  plane  containing  an  element  of  a  warped 

surface 75 

67.  Mutually  tangent  warped  surfaces 75 

68.  Divisions  of  warped  surfaces 76 


Xll  TABLE   OF   CONTENTS. 

ARTICLE.  PAGE 

69.  Infinite  directrices 70 

70.  But  one  infinite  directrix 76 

71.  Special  group  of  warped  surfaces 77 

72.  Orders  of  warped  surfaces 77 

73.  Higher  orders 77 

74.  Screw  surfaces ,77 


INTRODUCTION. 


How  can  a  solid  having  three  dimensions  be  exactly  repre- 
sented upon  a  surface  having  but  two  dimensions  ? 

This  is  the  problem  which  Descriptive  Geometry  seeks  to 
answer.  As  the  theoretical  basis  of  its  answer  it  develops  cer- 
tain laws  of  relationship  which  connect  the  figure  in  space  with 
its  expression  in  a  plane.  These  laws  belong  to  Protective 
Geometry  and  are  rigorously  mathematical ;  when,  however, 
actual  representations  of  real  objects  are  attempted,  the  results 
will  be  approximations  of  varying  degrees  of  accuracy  accord- 
ing to  the  skill  of  the  artist.  Descriptive  Geometry  is  an  art 
when  it  exercises  a  student  in  its  methods  ;  a  science,  when  it 
reveals  a  strictly  mathematical  basis  for  its  methods. 

To  the  technologist,  as  the  architect  or  mechanic,  it  is  not 
only  necessary  that  the  representation  should  be  derived  from 
the  original  and  suggest  it  in  a  general  way,  but  it  is  even 
more  imperative  that  the  original  itself,  which  may  have  been  a 
material  object  or  only  a  creation  of  the  imagination,  may  be 
reproduced  by  the  skilled  workman  with  the  aid  of  the  repre- 
sentation in  tangible,  material  form,  in  every  smallest  detail  of 
shape  and  measurement.  Because  rectangular  or  orthographic 
projection  accomplishes  this  twofold  object  best,  it  has  gen- 
erally been  allowed  to  usurp  the  whole  domain  of  Descriptive 
Geometry,  and  it  is  not  the  purpose  of  this  little  book  to  depart 
greatly  from  the  usual  though  inadequate  interpretation  of  the 
science.  For  the  sake  of  special  descriptive  properties  easily 
understood,  the  more  general  science  of  Projective  Geometry 
is  drawn  upon  for  a  few  isolated  propositions. 

A  knowledge  of  plane  and  solid  elementary  or  ancient  Geom- 
etry is  assumed.  Especial  attention,  however,  is  directed  to 


Z  DESCRIPTIVE   GEOMETRY. 

those  propositions  which  in  most  American  manuals  are  em- 
braced in  the  first  two  books  of  solid  Geometry.  The  student 
is  also  supposed  to  have  a  fair  knowledge  of  elementary  Al- 
gebra and  Trigonometry;  and  in  that  portion  of  this  work 
which  treats  of  figures  of  a  higher  order  than  the  first,  of 
elementary  Analytic  Geometry. 

When  statements  without  demonstration  are  made,  it  is  with 
the  expectation  that  the  student  will  think  them  through  and 
satisfy  himself  that  they  are  founded  in  formal  Geometry  or  in 
common  sense.  He  should  hold  in  mind  the  figures  presented 
to  him  for  consideration  until  in  imagination  he  can  see  them 
in  their  true  forms  and  relations.  He  will  then  be  able  to 
perform  operations  upon  them  as  upon  material  objects  pre- 
sented to  the  senses.  This  is  the  essence  of  Geometry.  No 
exercise  of  the  mathematical  faculties  can  be  more  productive 
of  beneficial  results,  and  there  is  no  other  that  will  give  a  more 
pleasurable  feeling  of  mastery. 

Preparatory  to  the  solution  of  the  problems  let  us  now  consider 
p  in  a  practical  way  the  meaning  of  ortho- 

graphic projection  and  discuss  briefly  the 
transformations  usually  made.  For  this 
purpose  let  us  take  a  piece  of  stiff  writing- 
paper  about  eight  inches  long  and  four 
wide  and  cut  it  as  in  Fig.  a,  —  two  slits 
along  the  medial  line  a&,  about  one  inch 
into  each  side. 

Take  another  piece  of  paper  of  the  same 
size  cutting  it  as  in  Fig.  /?,  —  a  two-inch 
slit  in  the  line  a&,  leaving  an  inch  on  each 
side  of  the  paper  uncut.  By  folding  over 
the  top  of  a  without  creasing  it  we  will  thrust  it  through  the 
slit  of  ft.  The  two  pieces  will  then  hinge  along  their  medial 
lines  and  may  be  made  to  assume  any  angle  with  each  other. 
We  will  place  them  so  that  their  dihedral  angle  is  about  90°. 
Make  one  of  the  pieces  as  nearly  as  possible  horizontal.  We 
will  call  it  the  horizontal  plane  of  projection  and  designate  it 


INTRODUCTION.  3 

by  H.  The  other  piece  will  be  approximately  vertical  in  posi- 
tion. We  will  call  it  the  vertical  plane  of  projection  and  desig- 
nate it  by  V.  Hold  the  papers  so  that  H  is  lower  than  the  eye 
and  F  in  front  of  it.  The  angular  space  FlG  g 

in  view  we  will  call  the  first  quadrant  and 
designate  it  by  1  Q  ;  the  angular  space  be- 
hind 1  Q  we  will  call  the  second  quadrant 
or  2  Q  ;  that  below  2  Q  will  be  3  Q  ;  that 
in  front  of  3  Q  and  below  1  Q  will  be  4  Q. 
The  medial  line  of  H  and  F,  or  the  hinge, 
we  will  call  the  ground  line  and  designate 
it  by  G. 

Suppose  a  point  situated  in  space  some- 
where in  1  Q,  and  perpendicular  lines  drawn 
from  the  point  to  H  and  F  respectively. 
These  lines  are  the  projecting  lines  of  the  point,  and  the  points 
in  which  they  pierce  H  and  F  are  respectively  the  H  and  F  pro- 
jections of  the  point.  Assuming  our  pieces  of  paper  to  be  true 
planes  at  right  angles  to  each  other,  the  two  projecting  lines 
determine  a  third  plane  perpendicular  to  both  H  and  F,  there- 
fore to  their  intersection  G.  This  third  plane  will  cut  from  H 
and  F  two  lines  each  of  which  is  perpendicular  to  G  at  the 
same  point.  If  now  we  reverse  our  operation  and  erect  perpen- 
diculars to  H  and  F  at  the  projections  of  a  point,  it  is  evident 
that  they  must  meet  in  space  and  determine  the  original  point. 
A  point  is  therefore  fully  determined  by  its  two  projections. 

When  we  revolve  H  and  F  upon  G  it  is  evident  that  the 
relation  of  the  intersections  of  the  third  plane  to  G  is  not 
changed.  When,  therefore,  H  and  F  form  one  continuous 
plane,  the  two  intersections  form  one  and  the  same  right-line  per- 
pendicular to  G.  We  thus  establish  the  important  proposition 
that  when  the  planes  of  projection  are  brought  into  coincidence 
the  right-line  joining  the  two  projections  of  any  point  is  and 
must  be  perpendicular  to  the  ground  line. 

If  instead  of  a  single  point  in  space  we  had  taken  a  system 
of  points  lying  in  a  right-line,  the  projecting  lines  of  all  these 


4  DESCRIPTIVE  GEOMETRY. 

points  would  lie  in  a  projecting  plane  whose  intersection  with 
H  or  V  would  form  the  H  or  V projection  of  the  line.  Ex- 
cept in  special  positions  to  be  studied  hereafter  the  H  and  V 
projections  of  a  line  determine  two  projecting  planes  whose 
intersection  in  space  is  the  original  line. 

Going  a  step  further  we  see  that  the  locus  of  all  the  projecting 
lines  to  H  or  V  of  all  the  points  of  a  space  curve :  is  a  continuous 
surface  generated  by  a  right-line  moving  along  the  curve  with 
all  its  positions  parallel,  is  a  cylindrical  surface  therefore,  and 
in  this  case  a  projecting  cylindrical  surface  whose  intersection 
with  its  plane  of  projection  is  the  like-named  projection  of  the 
space  curve.  The  horizontal  and  vertical  projecting  surfaces 
intersect  in  space  in  the  original  curve.  Hence,  in  general,  all 
the  points  of  any  solid  in  space,  therefore  the  solid  itself,  are 
fully  determined  by  their  H  and  V  projections. 

In  order  now  that  this  perfect  representation  of  a  figure  in 
space  may  be  had  upon  a  single  plane  surface,  we  first  conceive 
our  planes  of  projection  at  right-angles  to  each  other.  We  then 
suppose  a  revolution  upon  G  which  brings  them  into  coincidence 
and  perform  the  operations  which  this  revolution  necessitates. 
H  and  Fare  always  made  coincident  by  enlarging  the  angle 
between  upper  V  and  front  H  from  one  of  90°  to  one  of  180°. 

As  a  preliminary  exercise  let  us  discuss  the  first  part  of 
problem  3.  We  will  represent,  by  the  method  explained  above, 
a  line  a b  crossing  the  first  quadrant.  We  will  also  find  its  true 
length  between  H  and  F,  and  the  angles  it  makes  with  these 
planes.  Recurring  to  our  two  pieces  of  paper,  it  is  evident  that 
if  the  line  is  to  cross  1  Q,  the  point  in  which  it  pierces  Fmust  be 
above  6r,  and  that  in  which  it  pierces  Hiu  front  of  G.  With  these 
limitations,  assume  the  points  of  piercing  anywhere.  Call  the 
point  in  which  ob  pierces  H,h',  in  which  it  pierces  F,  v.  h  is  its 
own  H  projection  ;  the  H  projection  of  v  is  a  point  in  G  found  by 
drawing  a  perpendicular  from  v  to  G.  We  thus  know  two  points 
in  the  jET projection  of  a&,  therefore  the  ^projection  itself. 

Similarly  we  find  the  Fprojection.     The  former  we  designate 
by  a'&',  the  latter  by  a"  6",  as  in  Fig.  y. 
1  Part  III.  Art.  27. 


INTRODUCTION.  5 

To  find  the  true  length,  i,  of  ab  between  h  and  v,  we  can  use 
a'b'  from  G  to  h  as  the  leg  of  a  right-angled  triangle  whose  other 
leg  is  the  perpendicular  from  v  to  G.  The  hypothenuse  of  this 
triangle  is  the  length  sought.  A  convenient  construction  is 
given  in  Fig.  y.  The  basal  angle  between  a'b'  and  aubu  is  a', 
the  inclination  of  ab  to  H.  The  same  triangle  is  sometimes 
more  conveniently  constructed  in  V.  In  this  case  (a)  (6) ,  as 
shown  in  Fig.  y,  is  the  hypothenuse  and  the  length  along  this 
line  from  v  to  G  is  L.  This  method  may  be  explained  by 
saying  that  the  line  ab  is  revolved  about  some  point  in  its  H 
projection  until  it  is  parallel  to  V  when  its  V  projection  is  its 
true  magnitude. 


FIG.  7. 

The  process  here  illustrated  in  on6u  of  laying  a  plane  figure 
over  into  H  or  Vis  called  revolution.  The  relation  of  the  parts 
of  the  figure  in  space  to  each  other  is  not  disturbed,  while  its 
H  or  ^projection  is  taken  as  the  axis  of  revolution.  When 
some  other  line  in  H  or  V  is  taken  as  an  axis  the  figure  is  said 
to  be  developed. 

As  a  rule,  points  and  right-lines  are  revolved,  while  points, 
right-lines,  planes,  and  developable  surfaces  are  developed. 

When  a  plane  figure  lies  in  a  plane  perpendicular  to  (2,  or  in 
a  plane  making  a  very  small  angle  with  6r,  a  third  plane  of 


6  DESCRIPTIVE   GEOMETRY. 

projection  is  often  necessary.  When  used  it  is  generally  taken 
perpendicular  to  G  and  therefore  to  H  and  F. 

To  recapitulate  briefly. 

The  planes  of  projection  are  three,  each  one  perpendicular  to 
the  other  two;  one,  parallel  to  the  horizon  —  the  first  or  hori- 
zontal plane  ;  one,  perpendicular  to  this  and  extending  from  left 
to  right  —  the  second  or  vertical  plane  ;  one,  upon  the  right  ex- 
tending from  front  to  rear  —  the  third  or  perpendicular  plane, 
corresponding  to  what  is  known  in  architecture  as  plan,  front 
and  side  respectively. 

The  intersection  of  the  first  with  the  second  plane  is  called 
the  first  ground  line  or  simply  the  ground  line  ;  the  intersection 
of  the  second  with  the  third  is  taken  as  the  second  ground  line ; 
the  intersection  of  the  third  with  the  first  as  the  third  ground 
line.  The  first  and  second  planes  are  revolved  about  the  first 
ground  line  into  one  plane,  which  therefore  represents  both ; 
below  the  ground  line  is  front  horizontal  and  lower  vertical ; 
above  the  ground  line  is  back  horizontal  and  upper  vertical. 
The  third  plane  of  projection  is  revolved,  the  front  part  to  the 
right,  about  the  second  ground  line  until  it  coincides  with  the 
second  plane. 

These  transformations  are  further  illustrated  in  figures  1  and 
2,  3  and  4  ;  figures  1  and  3  show  the  three  planes  in  space 
marked  H,  V,  P,  before  revolution  and  the  first,  second,  third 
ground  lines  ;  figures  2  and  4  show  the  same  lines  and  planes 
transformed. 

Projection  has  the  same  meaning  as  in  elementary  Geometry 
and  is  effected  by  projecting  lines,  planes,  and  cylindrical  sur- 
faces ;  the  intersections  of  these  with  the  planes  of  projection 
are  the  projections  of  a  figure. 

After  reading  through  the  remaining  portion  of  the  Introduc- 
tion, the  student  should  proceed  to  solve  the  problems  with 
the  least  possible  help  from  Part  II. 

The  projections  of  given  lines,  or  the  traces  of  given  planes, 
are  represented  by  an  unbroken  line : 


INTRODUCTION.  7 

Whether  the  unbroken  line  belongs  to  the  former  or  latter  is 
shown  in  two  ways :  by  the  lettering,  and  by  the  fact  that  the 
traces  of  a  plane  always  meet  on  6r,  while  in  general  the  pro- 
jections of  a  line  do  not. 

When  the  continuation  of  a  projection  of  a  given  line  is  cov- 
ered by  a  plane  of  projection  or  by  a  solid  in  space,  the  covered 
portion  is  represented  by  short  dashes  : 


The  continuation  of  the  traces  of  a  given  plane  under  similar 
conditions,  by  dashes  nearly  twice  as  long : 


The  projections  of  a  required  line,  or  the  traces  of  a  required 
plane,  by  a  line  broken  at  much  longer  intervals  : 


while  the  covered  portions  of  both  are  represented  as  in  the 
given  line  and  plane. 

The  projections  of  an  auxiliary  line  are  alternate  dashes  and 
dots: 


The  traces  of  auxiliary  planes  are  longer  dashes  alternating 
with  two  dots : 


In  the  last  two  cases  no  difference  is  made  in  the  covered 
portions,  except  in  a  few  problems  where  a  distinction  seems 
necessary,  when  the  covered  projections  and  traces  are  drawn 
lighter.  Given  or  required  arcs  or  circles  are  drawn  full,  auxil- 
iary are  broken. 

A  series  of  points  is  used  for  connecting  the  projections  of 
points,  whether  in  space  or  developed. 

Developed  lines  and  traces,  when  neither  given  nor  required, 


8  DESCRIPTIVE   GEOMETKY. 

are  represented  as  auxiliary ;  for,  while  they  are  neither  new 
nor  in  a  measure  independent  figures,  they  are  helps  in  reaching 
the  result  sought. 

H,       means  the  horizontal  plane  of  projection. 

F,        the  vertical  plane  of  projection. 

P,        the  perpendicular  plane  of  projection. 

6r,        the  first  ground  line. 

6r2,       the  second  ground  line. 

Cr3,       the  third  ground  line. 

1 Q,      the  first  quadrant. 

2Q,      the  second  quadrant ;  etc. 

gn.,      given. 

rq.,  required;  that  is,  when  not  otherwise  stated,  the  first 
and  second  projections  are  required. 

pr.,       projection;  prs.,  projections. 

JT-pr.,the  horizontal  projection,  or  the  projection  in  the  first 
plane. 

F-pr.,  the  second  projection. 

P-pr.,  the  third  projection. 

7i,         the  point  in  which  a  line  pierces  the  first  plane. 

v,          the  point  in  which  a  line  pierces  the  second  plane. 

p,         the  point  in  which  a  line  pierces  the  third  plane. 

L,        the  true  length  of  a  line  between  h  and  v. 

line,  right-line,  unless  otherwise  qualified,  or  shown  by  the 
context  to  mean  line  in  general. 

pt.,       point;  pts.,  points. 

revolved  position,  means  a  position  assumed  when  a  space  figure 
is  revolved  into  a  plane  of  projection  on  a  line  of  this 
plane  through  the  corresponding  projection  of  the  fig- 
ure as  an  axis.  In  Fig.  y,  a^  is  a  rev'd  pos.  of  ab. 

developed  position,  the  same  as  above,  except  the  axis  of  revolu- 
tion is  any  other  line  of  the  corresponding  plane  of  pro- 
jection ;  in  general,  such  axis  will  be  the  corresponding 
trace  of  some  oblique  plane  which  contains  the  figure 
in  space.  In  Fig.  13,  (a)  (b)  is  dev.  pos.  of  ab. 


INTRODUCTION.  9 

ic,  ?/,  z,  ordinates  :  when  a  figure  is  given  by  its  x,  y,  z  ordinates, 
x  means  distance  from  P  parallel  to  G ;  ?/,  distance 
from  ^parallel  to  G3 ;  z,  distance  from  H  parallel  to 
6r2.  Positive  x  is  reckoned  from  P  to  the  observer's 
right ;  positive  y,  from  V  forwards ;  positive  z,  from 
H  upwards.  When  the  position  of  a  figure  is  given 
by  its  #,  y,  z  ordinates  the  position  of  P  is  shifted  from 
the  observer's  right  to  his  left,  to  correspond  with  the 
usual  assumptions  of  Analytic  Geometry. 

a&c,  etc.,          designates  the  figure  in  space. 

a'6'c',  etc.,        its  first  projection. 

a"6"c",  etc.,     its  second  projection. 

a'"b'"c"')  etc.,  its  third  projection. 

aAcj,  etc.,       the  figure  revolved  into  H. 

a2b2c2,  etc.,       the  figure  revolved  into  V. 

a363c3,  etc.,  the  figure  revolved  into  P.  When  a  figure  is 
revolved  into  any  plane  more  than  once,  double  sub- 
scripts should  be  used ;  as,  a12. 

(a),  (6),  (c),  etc.,  a  figure  developed  in  any  plane  of  pro- 
jection. When  several  times  developed  in  the  same 
plane  of  projection  the  figures  may  be  distinguished 
by  the  Arabic  numerals  ;  as,  (a)^  (a)2,  etc. 

t'Tt",  or  a&,  ccZ,  a  plane  in  space;  the  former  given  by  its 
traces,  the  latter  by  two  parallel  or  intersecting 
lines. 

t'T,      the  first  trace  of  the  plane  t'Tt". 

Tt",     the  second  trace. 

T"t'",  the  third  trace. 

K,        the  angle  which  the  first  and  second  traces  make  in  space. 

a',  the  inclination  of  a  line  to  H,  or  the  minimum  acute 
angle  which  it  makes  with  any  line  of  H  through  its 
foot. 

a",        the  inclination  of  a  line  to  V. 

a'",       the  inclination  of  a  line  to  P. 

e',  the  angle  of  inclination  of  a  plane  to  H,  or  the  plane 
angle  formed  by  drawing  two  lines,  one  in  the  plane 


10  DESCRIPTIVE   GEOMETRY. 

and  one  in  H,  each  perpendicular  to  the  £T-trace  of 

the  plane  at  the  same  point. 
e",         the  angle  of  inclination  of  a  plane  to  V. 
e'",       the  angle  of  inclination  of  a  plane  to  P. 
_L,        perpendicular. 
|[ ,         parallel. 
=,        equal. 
A,        triangle. 
O,       parallelogram. 
£,        a  given  angle  or  angle  in  general. 
]£,         a  required  angle. 
>,        greater  than. 
<,        less  than, 
windschief,  two  lines  crossing  each  other  in  space  but  not 

intersecting, 
oo,         infinity. 
Xj         given  point. 
Q?         required  point. 


DESCEIPTIVE  GEOMETEY, 

PART   I. 

EXEEOISES  AND  PKOBLEMS, 

In  the  following  exercises  and  problems  the  student  is  referred 
to  the  Introduction  for  explanations  of  abbreviations  and 
notation  ;  to  Part  II.  for  suggestions,  analyses  and  demonstra- 
tions ;  and  to  Part  III.  for  a  synopsis  of  the  subject-matter  of 
Descriptive  Geometry  as  covered  by  this  handbook. 

SECTION  I. 
Progressive  Course  on  the  Point,  Line   and  Plane. 

1.  Construct  the  projections  of  a  point,  a,  in  each  of  the 
four  quadrants. 

2.  Find  and  construct  five  other  positions  of  a  point,  a,  the 
first  of  which  shall  lie  in  the  fore  part  of  the  horizontal  plane. 

3.  Construct  four  lines  :  one,  a&,  crossing  the  first  quadrant ; 
one,  cd,  crossing  the  second  quadrant ;   etc.     Find  their  hori- 
zontal and  vertical  traces,  and  designate  the  former  by  h,  the 
latter  by  v. 

4.  Construct  four  lines,  a&,  cd,  etc.,  parallel  to  the  ground 
line,  one  in  each  quadrant. 

5.  Construct  a  line,  a&,  parallel  to  the  vertical  plane,  and 
inclined  to  the  horizontal  plane. 

6.  Construct  a  line,  06,  lying  in  the  horizontal  or  vertical 
plane ;  also  a  line,  cd,  lying  in  the  bisecting  plane  of  the  first 
and  third  quadrants,  and  a  line,  e/,  lying  in  the  bisecting  plane 
of  the  second  and  fourth  quadrants. 


12  DESCRIPTIVE   GEOMETRY. 

7.  Construct  lines,  a&,  cd,  etc.,  crossing  the  four  quadrants 
and  lying  in  planes  perpendicular  to  the  ground  line.     Use  the 
perpendicular  plane  in  each  case  as  a  third  plane  of  projection. 
Revolve  the  latter  upon  its  intersection  with  the  vertical  plane 
considered  as  a  second  ground  line,  and  show  the  true  position 
of  each  third  projection,  a'"b'",  c'"d'",  etc. 

8.  Find  and  represent  in  a  similar  way  six  other  positions  of 
lines  lying  in  planes  perpendicular  to  the  ground  line. 

NOTE.    From  this  point  on,  the  notation  and  abbreviations  explained  in 
the  Introduction  will  be  fully  used. 

9.  Construct  two  lines,  a&,  cd,  intersecting  in  the  pt.  a,  one 
II  to  F,  the  other  to  H. 

10.  Construct  any  two  lines,  a&,  cd,  intersecting  in  the  pt. 
a?,  2/,  z  ;  x  =  3,  y  =  2,  z  =  4.     Assume  origin  in  G. 

11.  Construct  a  cone  of  revolution, 

a)  with  base,  _B,  in  H  and  vertex,  $,  in  space ; 

b)  with  B  in  V  and  S  in  H. 

Show  in  a)  the  inclination,  a',  of  elements  to  H\  in  &),  the 
inclination,  a",  of  elements  to  F 

12.  Construct  a  cylinder  of  revolution, 

a)  with  B  in  H  and  axis  in  F; 

b)  with  B  in  F  and  axis  J_  to  F. 

13.  Gn.  a  line,  ab  ;  rq.  a',  a",  and  true  length,  L,  between 
its  traces,  ft,  v.     Solve  at  least  for  lines  crossing  1  Q  and  2  Q. 

14.  Gn.  two  pts.  in  space,  a  in  1  Q,  6  in  4  Q  ;   rq.  their  true 
distance. 

15.  Of  a  line,  a&,  in  space,  gn.  a'&',  h  and  a' ;  rq.  a"b"  and  -y. 

16.  Of  a  line,  a&,  gn.  a'6',  h  and  a"  ;  rq.  a"&",  a'  and  v. 

17.  Of  a  line,  a&,  gn.  a'6',  L  and  a';  rq.  a"6",  a",  v  and  7i. 
Solve  at  least  for  1  Q  and  2  Q. 

18.  Of  a  line,  a&,  gn.  a"6",  v  and  a' ;  rq.  a'b'  and  7i. 

19.  Of  a  line,  ab,  gn.  a"6",  v  and  a";  rq.  a'b'  and  h. 

20.  Of  a  line,  ab,  gn.  ft,  a'  and  a"  ;  rq.  a'6'  and  a"b".      Con- 
struct all  the  possible  positions  of  the  required  line  and  find  the 
'"4.  a'",  made  with  the  third  plane  of  projection,  P. 


EXERCISES   AND  PROBLEMS.  13 

21.  Gn.  a  line,  ab ;  rq.  the  pt.  p  of  ab  equally  distant  from 
H  and  V.     Solve  by  P. 

22.  Gn.  a  plane,  *W,  and  a"b"  of  a  line  therein  ;  rq.  a'b'  of 
the  line,  ab,  of  the  plane. 

23.  Gn.  a  plane,  t'Tt",  and  a'  of  a  pt.,  a,  therein  ;  rq.  a". 

24.  Gn.  a  plane,  t'Tt" ;  rq.  its  inclinations  to  .fl"  and  F,  ef 
and  e". 

25.  Gn.  a  plane,  t'Tt" ;  rq.  the  £  /f  which  the  traces  of  a 
plane  make  in  space.     See  Fig.  6. 

26.  Of  a  plane,  t'Tt",  gn.  t'T  and  K',  rq.  Tt". 

27.  Of  a  plane,  t'Tt",  gn.  K,  e'  and  a  pt.,  p,  in  «T;  rq.  £'T 
and  Tt". 

28.  Of  a  plane,  t'Tt",  gn.  Z'Tand  e;  rq.  Tt". 

29.  Of  a  plane,  t'Tt",  gn.  *Tand  e" ;  rq.  Tt". 

30.  Of  a  plane,  t'Tt",  gn.  e',  e"  and  a  pt.,  p,  in  t'T;   rq.  J'T 
and  a7*". 

31.  Gn.  a  plane,  t'Tt",  and  a  pt.,  p,  therein;  rq.  the  devel- 
oped position,  (p),  of  the  pt.  when  the  gn.  plane  is  developed 
into  Hon  t'T  or  into  Fon  Tt". 

32.  Gn.  a  plane,  t'Tt",  and  the  developed  position,  (p),  of  a 
pt.,  p,  therein  ;  rq.  p'  andp". 

33.  Of  a  pt.,  p,  gn.  p',  p"  and  (p)  ;  rq.  the  plane,  t'Tt", 
containing  the  pt. 

34.  Gn.  e"  of  a  plane,  t'Tt",  containing  the  pt.  p,  andp'  and 
(p)  of  the  pt.  ;  rq.  t'T,  Tt"  and  e'. 

35.  Gn.  the  ^  K  of  a  plane,  t'Tt",  containing  the  pt.  p,  and 
p'  and  (p)  of  the  pt.  ;  rq.  t'T  and  Tt". 

36.  Gn.  a  plane,  t'Tt",  and  the  distances  m'  and  m"  ;  rq.  the  pt. 
p,  whose  distances  from  t'T  and  Tt"  are  m'  and  m",  respectively. 

37.  Gn.  a  plane,  t'Tt",  and  a  line,  ab,  therein  ;  rq.  the  devel- 
oped line,  (ab) ,  and  the  ^  s  ft  and  ft",  which  the  line  ab  makes 
with  t'T&ud  Tt",  respectively. 

38.  Gn.  two  intersecting  lines,  ab,  cd\  rq.  the  plane,  t'Tt", 
containing  them,  and  the  K  8  which  these  lines  make  in  space. 
Find  the  bisector,  pq,  of  the  %.  between  the  gn.  lines. 

39.  Gn.  a  plane,  t'Tt",  a  line,  ab,  therein,  a  pt.,  p,  in  ab  and 
the  ^  8;  rq.  the  line  cd,  lying  in  trTt",  passing  through  p,  and 
making  with  ab  the  %.  8. 


14  DESCRIPTIVE   GEOMETEY. 

40.  Gn.  two  II  lines,  ab,  cd ;  rq.  their  plane,  t'Tt",  and  their 
true  distance  apart,  D. 

41.  Gn.  three  pts.,  a,  6,  c ;  rq.  their  plane,  t'Tt",  and  their 
true  relation,  (a)  (b)  (c),  as  shown  in  H  by  revolving  on  t'T, 
and  in  F'by  revolving  on  Tt". 

42.  Gn.  aline,  ab,  and  a  pt.,  p,  without  it;    rq.   the  plane, 
t'Tt",  of  ab  and  p,  also  the  distance  from  p  to  ab,  found  by 
developing  the  latter  into  Fupon  Tt". 

43.  Gn.  a  line,  ab,  a  pt.,  p,  without  it  and  the  ^  /3  ;   rq.  the 
line,  cd,  lying  in  the  plane,  t'Tt",  which  contains  ab  andp  ;  cd 
passes  through  p  and  makes  with  ab  the  £  /?. 

44.  Gn.  a  plane,  t'Tt",  a  pt.,  p,  within  it  and  the  2£  8 ;  rq. 
the  line  cd  of  £'7V',  containing  p  and  making  with  5"  the  ^  8. 

45.  Gn.  a  line,  ab,  and  the  ^  8;  rq.  the  plane,  t'Tt",  con- 
taining ab  and  so  situated  that  t'T  makes  with  ab  the  ^  8. 

46.  Gn.  a  plane,  t'Tt",  and  a  pt.,  p,  without  it ;  rq.  the  dis- 
tance, D,  from^  to  t'Tt">    Solve  by  a  projecting  plane  contain- 
ing p  and  J_  to  t'  T. 

47.  Of  a  plane,  t'Tt",  gn.  t'T,  a  pt.,p,  without  the  plane  and 
the  distance,  D,  fromp  to  t'Tt"  ;  rq.  Tt".     Reverse  46. 

48.  Gn.  a  plane,  t'Tt",  and  a  line,  ab,  without  it ;.  rq.  the  pt., 
p,  in  which  ab  pierces  t'Tt". 

1)  Line  general : 

a)  Plane  general ; 

b)  Plane  II  to  G. 

2)  Line  in  plane  J_  to  G : 

a)  Plane  general ; 

b)  Plane  II  to  G. 

Solve  for  each  case.     In  1)  a)  make  K  acute,  in  2)  a)  make 
K  obtuse. 

49.  Gn.  a  plane,  t'Tt",  and  a  pt.,  p,  without  it;  rq.  the  dis- 
tance, D,  fromp  to  t'Tt".     Solve  by  drawing  a  _L  from  the  pt. 
to  the  plane  and  determining  its  foot  in  the  plane  and  its  true 
length  between  the  pt.  p  and  the  foot. 

50.  Gn.  a  line,  ab,  and  apt.,£>,  without  it ;  rq.  the  plane, 
t'Tt",  containing  p  and  _L  to  ab.     Also  find  the  distance,  D, 


EXERCISES  AND   PROBLEMS.  15 

from  p  to  ab  by  finding  the  length  of  the  line  from  p  to  the  pt. 
q  where  ab  pierces  t'Tt". 

51.  Gn.  a  plane,  t'Tt",  and  a  line,  ab,  without  it ;  rq.  the  X  8 
which  ab  makes  with  t'Tt".     Solve  by  a  J_,  ph,  from  any  pt.,p, 
of  ab  upon  *W.     The  £  between  ph  and  a&  ==  90°  —  8. 

52.  Gn.  a  line,  a&,  and  a  pt.,  p,  without  it ;  rq.  the  line,  pq, 
containing^)  and  intersecting  ab  at  rt.  angles.     This  is  the  ex- 
tension of  50.     In  one  of  the  two  problems,  50  and  52,  assume 
the  data  so  that  Kis  >  90°,  in  the  other  so  that  A"  is  <  90°. 

53.  Gn.  a  plane,  t'Tt",  a  pt.,  p,  without  it  and  the  2£s  8  and 
0  ;  rq.  a  line,  pm,  containing  p,  making  with  //  the  ^  8  and  with 
t'Tt"  the  £  0. 

54.  Gn.  two  intersecting  planes,  t'Tt11  and  r'Rr" ;  rq.  their 
line  of  intersection,  inn, 

a)  when  the  traces  intersect  within  the  limits  of  the 

drawing ; 
6)  when  the  traces  do  not  so  intersect. 

55.  Gn.  two  intersecting  planes,  t'Tt"  and  r'Rr" ;  rq.  the 
^  <£  between  them.     Bisect  the  %.  <£  by  a  third  plane,  s'Ss". 

56.  Gn.  two  II  planes,  t'Tt"  and  r'Br" ;  rq.  their   distance 
apart,  D. 

57.  Gn.  two  windschief  lines,  ab  and  cd  ;  rq.  the  plane,  t'Tt", 
containing  a b  and  II  to  cd. 

58.  Gn.  two  windschief  lines,  ab  and  cd,  and  the  pt.  p ;  rq. 
the  plane,  t'Tt",  containing^  and  II  to  ab  and  cd. 

59.  Gn.  two  windschief  lines,  ab  and  cd  ;  rq.  their  distance, 
D,  and  its  projections,  p'q'  a,udp"q". 

60.  Gn.  two  windschief  lines,  ab  and  cd,  and  the  pt.  p  ;  rq. 
the   line,  pq,  passing  through  p  and  cutting  both  ab  and  cd. 
Solve,      aj  with  the  traces  of  auxiliary  planes  ; 

b)  without  such  traces. 

61.  Gn.  a  plane,  t'Tt",  and  a  pt.,  p,  without  it ;  rq.  a  plane, 
r'Rr",  containing  p  and  II  to  t'Tt". 

62.  Gn.  a  plane,  t'Tt",  and  the  distance  D ;  rq.  a  plane, 
II  to  f-'Tfc"  and  at  the  distance  D. 


16  DESCRIPTIVE   GEOMETRY. 

63.  Gn.  a  plane,  t'Tt'1,  and  without  it  a  line,  ab,  and  a  pt., 
p  ;  rq.  a  plane,  r'Rr",  containing  p,  II  to  ab  and  _L  to  t'Tt". 

64.  Gn.  a  plane,  t'Tt",  a  line,  ab,  within  it  and  the  ^  8  ;  rq. 
a  plane,  r'JRr",  containing  ab  and  making  with  t'Tt"  the  ^  8. 
Construct  when  8  =  60°. 

65.  Gn.  a  plane,  t'Tt",  a  line,  ab,  without  it  and  the  ^  </>  ; 
rq.  a  plane,  r'Rr",  containing  ab  and  making  with  t'Tt"  the  ^  <£. 

66.  Gn.  a  plane,  t'Tt",  a  line,  ab,  within  it,  a  pt.,  p,  without 
it  and  the  2f.  8  ;  rq.  a  liuQ,pq,  connecting  p  and  ab   and  making 
with  t'Tt"  the  £  8. 

NOTE.    In  trihedrals  the  face  Ys  are  a,  &,  y,  and  the  opposite  dihedrals 
are  A,  B,  C,  respectively. 

67.  Gn.  a,  ft  y  ;  rq.  A,  B,  C. 

68.  Gn.  a,  ft  (7;  rq.  .4,  JB,  y. 

69.  Gn.  a,  ft  A]  rq.  B,  C,  y. 

70.  Gn.    ft  C,  A  ;  rq.  £,  y,  a. 

71.  Gn.  a,  C,  A  ;  rq.  _B,  ft  y. 

72.  Gn.  A,  B,  C  ',  rq.  a,  ft  y. 

73.  Gn.  a,  C,  ((3  +  y)  ;  rq.  .4,  5,  ft  y. 

74.  Gn.  a,  C,  (£  -  y)  ;  rq.  .4,  5,  0,  y. 

75.  Gn.  a  plane,  t'Tt",  and  a'b'c'd'e'  of  a  pentagon,  abcde, 
lying  therein  ;    rq.  a"b"c"d"e"  and  the  true  figure,   (a)  (b)  (c) 

(<*)(«)• 

76.  Of  a  plane  pentagon,  a&cdte,  gn.  a'b'c'd'e'  and  a"b"c"  ;  rq. 
d"e"  and  the  true  figure,  (a)  (b)  (c)  (d)  (e)  ,  without  constructing 
the  plane  of  the  pentagon. 

77.  Of  a  plane  rectangle,   abed,  gn.   a'b'c'd'  and  the  true 
length,  L,  of  the  side  ab  ;  one  vertex,  a,  lies  in  H\  rq.  a"b"c"d", 
the  plane,  t'Tt",  of  the  rectangle  and  the  true  figure,  (a)  (b) 


78.  Gn.  a  pyramid,  S-abc,  andp"  of  a  pt.,  p,  upon  its  sur- 
face ;  rq.  p'.     Assume  the  pyramid  with  its  base  in  //. 

79.  Gn.  a  pyramid,  S-abcd',  rq.  the  length,  L,  of  the  edge 
/Sa,  the  ^  a"  of  inclination  of  Sa  to  F,  the  ^  <£  between  the 
two  faces  Sab  and  £&c,  the  ^  e"  of  inclination  of  the  face  Scd 


EXERCISES  AND   PROBLEMS.  17 

to  V  and  the  development  of  the  pyramid  in  F,  (S)  (a)  (b) 
(c)  (d) ,  the  solid  being  opened  on  the  edge  Sc  and  the  devel- 
oped lateral  surface  remaining  attached  in  F  to  the  basal  edge 
ab.  Assume  the  pyramid  with  the  base  in  F. 

80.  Of  a  triangular  pyramid,  S-abc,  gn.  its  base,  abc,  in  H 
and  the  lengths  L^  L2,  L3,  of  the  edges  Sa,  Sb,  Sc,  respec- 
tively ;  rq.  the  projections  of  the  pyramid. 

81.  Of  a  triangular  pyramid,  S-abc,  gn.  its  base,  abc,  in  F, 
two  lateral  edges,  So,,  Sb,  and  its  altitude,  A ;  rq.  the  projec- 
jections  of  the  pyramid. 

82.  Of  a  triangular  pyramid,  S-abc,  gn.  the  face  ^s  a,  /?,  y, 
of  the  trihedral  at  the  vertex  S,  one  basal  edge,  ab,  and  the 
conditions  that  the  base  shall  lie  in  H  and  that  the  inclinations 
of  the  lateral  faces  to  If  shall  be  equal ;  rq.  the  projections  of 
the  pyramid. 

83.  Of  a  triangular  pyramid,  /S-abc,  gn.  the  face  ^s  a,  /?,  y, 
at  the  vertex  S,  two  lateral  edges,  Sa,  Sb,  and  the  %  e'  of 
inclination, 

a)  of  the  face  Sab  to  H\ 

b)  of  the  face  Sac  to  H-, 

rq.  the  projections  of  the  pyramid.     Assume  the  base  in  H. 

84.  Gn.  a  triangular  pyramid,  S-abc ;   rq.  the  circumscribed 
sphere  and  its  centre,  C. 

85.  Gn.  a  triangular  pyramid,  S-abc ;  rq.  the  inscribed  sphere 
and  its  centre,  C. 

86.  Gn.  an  edge,  ab,  in  H  of 

a)  a  regular  tetrahedron  ; 

b)  a  regular  octahedron  ; 

c)  a  regular  icosahedron  ; 

d)  a  regular  dodecahedron  ; 

rq.  the  projections  of  the  polyhedrons  and  their  development  in 
H.  ab  is  to  be  assumed  in  each  case  as  one  side  of  a  face 
lying  in  H,  upon  which  the  polyhedron  rests.  No  one  of  the 
sides  of  this  basal  face  is  to  be  taken  either  II  or  _L  to  G. 

87.  Gn.  an  edge,  ab,  of  a  regular  icosahedron  II  to  H  and  an 


18  DESCRIPTIVE   GEOMETRY. 

adjacent  edge,  ac,  making  ^s  of  30°  and  45°  with  H  and  V 
respectively  ;  rq.  the  projections  of  the  solid. 

88.  Gn.  a  plane,  t'Ti",  and  a  pyramid,  S-abc  ;  rq.  to  deter- 
mine whether  the  plane  cuts  the  pyramid  between  the  limits  of 
the  vertex  and  base. 

89.  Gn.  a  pyramid,  S-abc,  and  an  intersecting  plane,  t'Tt"  ; 
rq.  the  figure  of  intersection,  mno. 

90.  Gn.  a  prism,  abc-a^b^,  and  an  intersecting  plane,  t'Tt"  ; 
rq.  the  figure  of  intersection,  mno. 

91.  Of  the  hexagonal  pyramid,  S-abcdef,  gn.  the  base,  abcdef, 
in  //  and  the  lengths  L^,  L2,  L3,  of  So,,  Sc,  Se,  respectively  ; 
rq.  the  pyramid  and  its  intersection,  mno  etc.,  by  a  plane,  t'Tt") 
which  cuts  off  equal  distances  from  Sa,  Sc,  Se. 

92.  Gn.  a  line,  de,  and  a  pyramid,  S-abc  ;  rq.  the  pts.  m,  ?i, 
in  which  de  pierces  8-abc. 

93.  Gn.  a  line,  de,  and  a  prism,  abc-a^b^  ;  rq.  the  pts.  m,  n, 
of  intersection. 

94.  Gn.  two  intersecting  pyramids,  S-abcd  and  T-xyz,  with 
bases  in  H\  rq.  the  figure  of  intersection,  mno  etc. 

95.  Gn.  two  intersecting  prisms,  o6c<i-a161c1(f1  and  xyz-x^y^, 
with  bases  in  V\  rq.  the  figure  of  intersection,  mno  etc. 

96.  Gn.    a   prism,    abcd-afifad^   with   base    in  H  and   an 
intersecting  pyramid,  S-xyz,  with  base  in  V',  rq.  the  figure  of 
intersection,  mno  etc. 

97.  Gn.  unit  rectangular  axes,  ox,  oy,  oz,  II  to  the  intersec- 
tions of  HV,  HP,  VP,  respectively  ;  rq.  their  projections,  o^, 
°\y\i  °izn  upon  a  fourth  plane,  t'Tt",  which  is  inclined  to  H  at 
an  £  of  80°  and  whose  t'T  makes  with  G  the  £  30°,  also  rq. 
the  development  of  t'Tt"  on  t'T  in  H  and  the  projected  axes 


98.  Project  axonometrically  the  star-dodecahedron,  the  ratio 
of  unit  rectangular  axes  in  space  being  in  projection,  x  :  y  :  z 
=  -Qj.  :  i  :  1  ;  the  inclinations  of  the  projected  axes  to  G  being 
tan  £  xoG  =  ^y,  tan  ^  yoG  =  £,  tan  ^  zoG  =  GO. 

99.  Gn.  in  H  two  A,  a&c  and  mno  ;  rq.  the  projections  of 
abc  when  the  H  projection,  a'6'c',  is  similar  to  mno.     Let  the 


EXEKCISES    AND   PROBLEMS.  19 

vertex  c  remain   in  H  during   the   necessary  rotation  of   the 
plane  of  abc. 

100.  Gn.  the  H  projection,  a'b'c',  of  a  A  abc,  the  altitude  of 
the  vertex  b  and  another  A,  mno  ;  rq.  the  V  projection,  a"b"c", 
when  abc  is  similar  to  mno. 

101.  Of  the  three  concurrent  edges  of  a  cube,  OA,  OB,  0(7, 
the  V  projections,  0"A",  0"B",  of  two  are  known;  rq.  the  cor- 
responding projection,  0"C",  of  the  third. 

102.  Gn.  0"A"  as  in  101  and  the  directions  of  0"B"  and 
0"C"  ;  rq.  the  corresponding  lengths  of  the  two  last. 

103.  Gn.  0"A"  as  in  101  and  the  lengths,  L^  L2,  of  0"B", 
0"C" ;  rq.  the  corresponding  positions  of  the  two  last. 

104.  Gn.   0"A"  as   in  101,  the  direction  of  0"B"  and  the 
length,  L2,  of  0"C"  ;  rq.  the  corresponding  length  of  0"B"  and 
position  of  0"C". 

SECTION  II. 
Additional  Exercises  upon  the  Point,  Line  and  Plane. 

105.  Gn.  three  pts.,  a,  b,  c,  as  the  vertices  of  a  triangle,  abc, 
by  a',  b',  c',  and  the  distances  d1?  d2,  ds,  respectively  above  H\ 
rq.  a"b"c"  and  the  true  figure  of  the  A  abc. 

106.  In  a  plane,  t'Tt",  given  by  t'T  and  a  pt.  a,  lies  a  regu- 
lar octagon  with  a  for  its  centre  and  one  side,  be,  II  to  t'T;  rq. 
the  prs.  of  the  octagon. 

107.  Gn.  in  H  a  O,  abed,  and  the  distance  d ;  rq.  the  prs. 
of  abed  when  its  centre,  m,  is  at  the  distance  d  above  H  and 
a'b'c'd'  is  a  square. 

108.  Gn.  the  #-pr.,  a'b'c',  of  a  A  ;  rq.  the  F-pr.,  a"b"c", 
when  a&c  in  space  is  equilateral  and  the  vertex,  a,  lies  in  H. 

109.  Gn.  a  pt.,  o,  as  centre  of  a  regular  hexagon  and  the 
jff-pr.,  a'b',  of  one  side,  ab ;  rq.  the  hexagon,  abcdef,  and  its 
plane,  t'Tt". 

110.  Gn.  the  side,  ab,  of  a  square,  abed,  and  the  ^  S;  rq. 
the  square  and  its  plane,  t'Tt",  when   the  diagonal  ac  makes 
with  H  the      8. 


20  DESCRIPTIVE  GEOMETRY. 

111.  Gn.  two  pts.,  a,  6,  and  the  distance  d',  rq.  the  equilat- 
eral A  abc  when  c  is  at  the  distance  d  from  H. 

112.  Of  a  A,  abc,  there  are  given  the  vertices  a,  6,  the  H-pr. 
cf,  a  square,  K,  of  area  equal  to  that  of  abc ;  rq.  c",  the  plane, 
t'Tt",  of  the  A  abc  and  the  true  figure,  (a)  (b)  (c),  of  the  latter. 

113.  Gn.  a  pt.,  p,  a  line,  ab,  _L  to  If  and  a  line,  cd,  JL  to  F; 
rq.  the  line,  pq,  through  p  and  intersecting  ab  and  cd. 

114.  Gn.  three  lines,  ab,  cd,  e/;  rq.  Us,  a^,  c^,  e^,  to  the 
three  lines  respectively,  each  of  which  shall  cut  the  two  given 
lines  to  which  it  is  non-parallel. 

115.  Five  pts.,  a,  5,  c,  d,  e,  are  gn.  by  their  co-ordinates,  as 
follows : 

a          b          c          d          e 
x  =    0         30        48        40         10 
y  =  15         35  6         46         28 

z  =  48         10         30         48  0 

rq.  the  true  figure,  (a)  (b)  (c),  of  the  A  abc  and  its  plane,  t'Tt", 
the  distances,  d1?  d2,  of  d  and  e  from  tf'TY'  and  the  pt.  q  in 
which  the  line  de  pierces  t' Tt". 

116.  Gn.  a  pt.,p,  and  a  line,  ab ;  rq.  the  regular  decagon, 
mno  etc.,  whose  centre  is  p  and  one  of  whose  sides,  mn,  falls 
in  ab. 

117.  Gn.  a  plane,  £'7Y",  and  figure,  abc  etc.,  therein,  whose 
jBT-pr.,  a'b'c'  etc.,  is  a  regular  octagon  ;  rq.  a"b"c". 

118.  Gn.  a  pt.,  p,  and  two  lines,  ab  and  cd,  both  J_  to  $  ;  rq. 
the  line,  pq,  through  p  and  intersecting  ab  and  cd. 

119.  Given  a  plane,  t'Tt",  and  a  pt.,  p,  without  it ;  rq.  a  line, 
pq,  through  p  and  ||  to  t'Tt",  whose  prs.,p'g'  and  p"q",  shall 
be  ||  lines,  when  Fand  IT  are  made  to  coincide  as  usual. 

120.  Gn.  a  line,  ab,  and  two  planes,  t'Tt"  _L  to  H  and  r'JSr" 
_L  to  F;  rq.  the  pt.,  p,  of  ab  equally  distant  from  t'Tt"  and 
r'Rr". 

121.  Gn.  a  pt.,  p,  and  the  planes,  t'Tt"  and  r'Rr" ;  rq.  the 
line,  pg,  through  jp  ||  to  both  planes. 

122.  Gn.  a  line,  a&,  and  a  plane,  t'Tt"',  rq.  the  line,  jpg, 
of  m"  J_  toa&. 


EXERCISES   AND  PROBLEMS.  21 

123.  Gn.  a  pt.,  p,  and  two  windschief  lines,  ab  and  cd ;  rq. 
the  line,  pq,  passing  through  p,  cutting  ab  and  _L  to  cd  ;  or,  rq. 
the  line,  pt,  passing  through  p  and  _L  to  both  ab  and  cd. 

124.  Gn.  a  plane,  t'Tt",  and  two  lines,  ab  and  cd ;  rq.  the 
line,  j>g,  _L  to  Z'TZ"  and  intersecting  ab  and  cd. 

125.  Gn.  a  line,  a&,  and  the  2£  S ;  rq.  the  line,  pq,  of   H, 
which  makes  with  ab  the  ^  8. 

126.  Gn.  a  pt.,  p,  a  line,  a&,  and  the  ^  30°  ;  rq.  the  line,  pq, 
passing  through  p,  cutting  ab  and  making  with  Fthe  ^  30°. 

127.  Gn.  a  plane,  t'Tt",  the  lines  ab  and  cd,  and  the  dis- 
tance  d;  rq.  the   line,   pq,   ||  to  t'Tt"  at  the  distance  d  and 
intersecting  ab  and  ccL 

128.  Gn.  two  planes,  t'Tt"  and  r'Rr",  a  line,  a&,  and  ratio 
m,  ml;  rq.  the  pt.,  p,  of  a&  at  the  distances  q,  q1}  from  t'Tt"  and 
r'Rr"  respectively  :    gn.  g  :  #1  =  w>  '•  rtn>\> 

129.  Gn.  three  planes,  t'Tt",  r'Rr",  s'Ss",  and  the  distances 
m,  n,  q ;  rq.  the  pt.,  p,  at  the  distances  m,  w,  g,  from  the  three 
planes  respectively. 

130.  Gn.  two  pts.,  a,  b,  the  plane  t'Tt"  and  the  lengths  L^ 
Lz',  rq.  the  A,  abc,  constructed  upon  ab  as  a  base,  with  sides 
ac,  be,  respectively  equal  to  Lj,  L2,  and  with  the  vertex  c  in 
t'Tt". 

131.  Gn.  two  pts.,  a,  b,  a  line,  mn,  and  an  area,  A  ;  rq.  a  pt., 
c,  in  mn,  so  situated  that  the  A  abc  shall  have  the  area  A. 

132.  Gn.  a  line,  «6,  a  pt.,  p,  and  the  distance  c? ;  rq.  the 
plane,  t'Tt",  containing  ab  and  at  the  distance  d  from  p. 

133.  Gn.  three  pts.,  a,  6,  c,  no  one  of  which  lies  in  H or  F; 
rq.  a  pt.,  p,  in  H,  equally  distant  from  a,  b,  c. 

134.  Gn.  two  pts.,p,  q,  the  line  ab  and  the  distance  d;  rq. 
the  line,  qr,  through  q,  intersecting  ab  and  at  the  distance  d 
from  p. 

135.  Given  two  distances,  D1}  D2,  and 

a)  three  pts.,p,q,  r, 

6)  two  pts.,  p,  q,  and  a  line,  a&, 

c)  one  pt.,  p,  and  two  lines,  ab,  cd ; 


22  DESCRIPTIVE   GEOMETRY. 

rq.  a)  a  line,  pi,  through  p  at  the  distances  Dj,  D2,  from 

g,  r,  respectively, 
6)  a  line,  p£,  through  £>,  at  the  distance  Dl  from  g, 

and  Z>2  from  a&, 
c)  a  line,  pt,  through  p,  at  the  distances  D^  D2,  from 

ab,  cd,  respectively. 

136.  Gn.  four  pts.,  a,  b,  c,  d,  not  in  the  same  plane,  and  the 
distances  m,  n,  ^>,  g;  rq.  a  fifth  pt.,  a?,  whose  distances  from 
a,  6,  c,  d,  shall  be  as  m  :  n  :  p  :  g,  respectively. 

Take  a,  b,  c,  in  //.     How  many  solutions  ? 

137.  Gn.  four  pts.,  a,  b,  c,  d,  not  in  the  same  plane  and  the 
distances  m,  n,  p,  g;  rq.  a  plane,  t'Tt",  whose  distances  from 
a,  &,  c,  d,  shall  be  as  m:n:p:  g,  respectively. 

Take  a,  6,  c,  in  H.     How  many  solutions  ? 

138.  Gn.  three  pts.,  «,  6,  c,  and  the  distances  m,  n,£>;  rq.  a 
plane,  t'Tt",  whose  distances  from  a,  6,  c,  shall  be  m,  n,  |), 
respectively. 

How  many  solutions  ? 

139.  Gn.  four  pts.,  a,  6,  c,  d,  not  lying  in  the  same  plane 
and  the  quantities  m,  w,|),  g  ;  rq.  a  fifth  pt.,  x,  whose  distances 
from  the  planes  a&c,  abd,  acd,  bed,  shall  be 

a)  as  m:n'.pi  g, 
6)  equal. 

140.  Gn.  a  pt.,  ^>,  two  windschief  lines,  ab,  cd,  and  the  ^  s 
8  and  0 ;  rq.  a  line,  pq,  passing  through  p  and  making  with  ab, 
cd,  the  ^  s  8,  0,  respectively. 

141.  Gn.  a  pt.,  p,  two  planes,  Z'TV',  rW,  and  the  ^s  3',  8"  ; 
rq.  the  plane,  x'Xx",  making  with  t'Tt",  r'Er",  the  ^  s  8',  5", 
respectively. 

142.  Gn.  two  windschief  lines,  ab,  cd,  and  the  ^  s  /3',  ft"  ;  rq. 
a  line,  pq,  intersecting  ab,  cd,  and  making  with  them  the  ^s  /3', 
/?",  respectively. 

143.  Gn.  a  line,  ab,  and  two  pts.,  m,  n  ;  rq.  in  ab  the  pt.,  p, 
the  sum  or  difference  of  whose  distances  from  m  and  n  shall  be 
a  minimum  or  a  maximum  respectively. 

144.  Gn.  a  plane,  t'Tt",  and  two  windschief  lines,  ab.  aJ ;  rq. 


EXERCISES   AND   PROBLEMS.  '  23 

the  minimum   line,  pq,  which   is   II  to  t'Tt"  and  cuts  db  and 
od. 

145.  Gn.  a  plane  figure,  abc,  etc.,  of  n  sides,  whose  area  is 
A  ;  let  the  areas  of  its  projections  upon  H,  F,  P,  be  A^  A2,  A3, 
respectively  ;  rq.  to  prove  that  A2  =  A*  -\-  A22  -f-  A/. 

146.  Of  a  trihedral  angle,  gn.  a)  a,  /?,  (A  +  B  +  C),orb)  A, 
B,  (a  +  ft  +  y)  5  rq-  a)  y,  A,  B,  C,  6)  C,  a,  0,  y. 

147.  Of  a  trihedral  angle,  gn.  a,  the  £  «i  of  elevation  opposite, 
and  the  ^  e2  of  elevation  adjacent  to  the  face  of  a  ;  rq.  /?,  y,  and 
the  ^  e3  of  elevation. 

148.  Of  a  trihedral  angle,  gn.  a  and  the  adjacent  ^s  of  ele- 
vation e2,  e3  5  r<3-  P,  7«  ei- 

149.  Of  a  quadrahedral  2£,  S,  gn.  all  the  face  £s,  a,  /?,  y,  8, 
and  the  dihedral  £,  JL,  l}Ting  between  a  and  /?  ;  rq.  the  complete 
projection  of  the  quadrahedral  £  £. 

150.  Of  a  quadrahedral  ^,  6Y,  gn.  a,  /?,  y,  and  the  dihedrals 
(7,  Z>,  both  adjacent  to  the  unknown  face  £  8 ;  rq.  the  com- 
plete projection  of  the  quadrahedral  S. 

151.  Of  a  regular  tetrahedron,  abed,  gn.  the  vertices  a,  6,  and 
the  ^C  45°  as  the  inclination  which  the  line  cd,  joining  c  and  d, 
makes  with  H;  rq.  the  tetrahedron. 

152.  Given  three  concurrent  edges,  a&,  ac,  ad,  of  a  parallel- 
epiped and  the  face  ^s,  a,  /3,  y,  between  a&,  ac ;  ac,  ad ;  ad, 
a&,  respectively  ;  rq.  the  parallelepiped  abcdefgh. 

153.  Gn.  two  pts.,  a,  /*,  as  the  opposite  vertices  of  a  cube, 
abcdefgh,  and  the  2£  30°  as  the  inclination  of  the  edge  ab  to  H', 
rq.  the  cube. 

154.  Gn.  a  cube,  abcdefgh;  rq.  the  solid  m?ro  etc.,  formed 
by  passing  through  the  edges  of  the  cube  planes  each  J_  to  the 
diagonal  plane  in  which  the  edge  lies. 

155.  Gn.  in  H  a  A,  a'b'c',  as  the  JET-pr.  of  a  face  of  a  regular 
tetrahedron,  abed,  and  d,  the  distance  above  H  of  its  centre,  o; 
rq.  the  tetrahedron. 

156.  Gn.  two  pts.,  a,  b,  as  the  terminations  of  an  edge  of  a 
regular  icosahedron,  and  the  ^e ;  rq.  the  icosahedron  when  one 
of  its  faces,  a&c,  makes  with  H  the  ^.t. 


24  DESCRIPTIVE  GEOMETEY. 

157.  Gn.  three  pts.,  b,  c,  d,  as  the  terminations  of  three  con- 
current edges,  ab,  ac,  ad,  of  a  rectangular  parallelepiped  ;  rq. 
the  projections  of  the  parallelepiped  abcdefgh. 

158.  Of   a   hexagonal   pyramid,  S-abcdef,    gn.  the   regular 
hexagonal  base,  abcdef,  the  radius,  R,  of   the   circumscribed 
sphere,  the  altitude,  A,  of  the  pyramid  and  the  ^  fi  which  the 
side  Sa  makes  with  JJ;  a  plane,  t'Tt",  passes  through  the  centre 
o,  of  the  sphere,  cutting  from  the  edges  Sa,  Sc,  Se,  the  equal 
lengths  Sm,  Sn,  Sq ;  rq.  the  projections  of  the  truncated  pyra- 
mid abcdef-mxnyqz. 

159.  Of  a  quadrilateral  pyramid,  S-abcd,  gn.  the  base,  abed, 
in  H,  the  ^  e  which   the  face  Sab  makes  with  H,  the  ^  s  8^ 
82,    which  the  face  Sab  makes  with  the  faces  /Sac,  Sbd,   re- 
spectively, and  the  area   A  ;    rq.  the  pyramid   S-abcd   and  a 
plane  section,  mnop,  in  the  form  of  a  parallelogram  and  of  the 
area  A. 

160.  Of  a  pentagonal  truncated  prism,  a&cd'e-a1&1c1fZ1e1,  there 
is  given   in  H  the  base   abode,  the  face  aba^  and  the    ^  j3 
which  the  edges  make  with  H',  rq.  the  prism. 

161.  Of  a  tetrahedron,  abed,  gn.  the  edges  ab,  ac,  ad,  be,  bd, 
and  the  radius,  R,  of  the  circumscribed  sphere,  also  an  area, 
A',  rq.  the  tetrahedron,  also  a  section,  mnpq,  in  the  form  of  a 
parallelogram  and  of  the  area  A. 

162.  Of  a  regular  pentagonal  pyramid,  /S-abcde,  gn.  the  alti- 
tude, A,  the  length,  L,  of  the  basal  edge  ab,  the  plane  of  the 
base,  t'Tt",  which  makes  with  H  the  £  30°,  and  whose  //  trace, 
i'T,  makes  with  G  the  2£  45°,  ab  lies  in  £'T;  rq.  the  prism. 

163.  Of  a  hexagonal  pyramid,  S-abcdef,  gn.  the  base  abcdef 
in  H,  the  length,  L,  of  the  edge  Sa  and  the  ^s  a,  /?,  which 
the    edges    Sc,    Se,    respectively     make    with    H;     rq.    the 
pyramid. 

164.  Of  a  quadrilateral  pyramid,   S-abcd,  with  base  in  H, 
gn.  the  faces  Sab,  Sad,  the  2£S  between  them  and  the  2£s  ex, 
€2,  which  the  faces  Sbc,  Scd,  make  respectively  with  H',  rq.  the 
pyramid  and  a  section,  mno,  made  by  a  plane,  t'Tt",  equally 
inclined  to  Sab,  Sad  and  H. 


EXERCISES   AND  PROBLEMS.  25 

165.  Gn.  a  pt.,  a,  as  a  vertex  of  a  cube,  abcdefgh,  the  length, 
L,  of  an  edge  of  the  cube  and  a  A,  mno,  in  H;  rq.  the  prs.  of 
the  cube  when  the  sides  of  the  JJ-pr.  are  ||  to  mno. 

166.  Gn.  a  pt.,  S,  passing  through  $,  the  lines  $w,  Sn,  Sp., 
and  the  lengths  L^  L2 ;  rq.  a  pyramid,  S-abcd,  with  S  for  its 
vertex,  with  three  lateral  edges,  $a,  $6,  $c,  lying  in  Sm,  Sn,  /Sp, 
respectively,  with  the  lateral  edges  Sa,  Sb,  equal  to  LI,  L2, 
respectively,  and  with  the  base  a  rectangle,  abed. 

167.  Gn.  a  regular  dodecahedron  and  an  intersecting  plane, 
t'Tt" ;  rq.  the  figure  of  intersection,  mno  etc. 

168.  Gn.  an  octahedron,  abcdef,  and  an  intersecting  tetra- 
hedron, ghik ;  rq.  the  figure  of  intersection,  mno  etc. 

169.  Gn.  a  regular  dodecahedron  and  a  concentric  intersect- 
ing cube ;  rq.  the  figure  of  intersection,  mno  etc. 

170.  Gn.  a  cube  ;  rq.  the  rhombic  dodecahedron  formed  by 
truncating  each  edge  with  a  plane  passing  through  the  centres 
of  the  four  adjacent  edges.     Project  the  new  solid,  axonomet- 
rically,  as  given  in  98. 

SECTION  III. 

Progressive  Course  on  Lines  and  Surfaces  of  an  Order  higher 
than  the  first. 

171.  Given  a  parallelogram,  abed ;  rq.  by  pts., 

a)  the  inscribed  ellipse, 

6)  the  hyperbola  with  the  diagonals  of  the  O  as  asymp- 
totes and  tangent  to  two  opposite  sides.  Con- 
struct the  principal  axes. 

c)  the  parabola,  tangent  to  ab  at  its  middle  pt.,  o,  and 
passing  through  the  pts.,  cd.     Find  the  principal 
axis,  007,  and  the  tangent  at  its  vertex,  oy. 
Make  three  different  UJ  for  a),  6),  c). 

172.  a)  Construct  the  ellipse  as  the  envelope  of  tangents 

drawn  in  a  gn.  O,  abed. 

b)  Construct  the  hyperbola  as  the  envelope  of  tangents 

drawn  between  two  gn.  intersecting  lines,  a&,  cd, 
used  as  asymptotes. 


26  DESCRIPTIVE    GEOMETRY. 

c)  Construct  the  parabola  as  the  envelope  of  tangents 
drawn  between  two  gn.  intersecting  lines,  ab,  cd. 

173.  Gn.  the  principal  axes,  AA',  BB',  of  an  ellipse,  or  any 
two  conjugate    diameters,  aa',  {3(3* ;   rq.  the   ellipse  by  circles 
about  the  principal  axes  or  conjugate  diameters. 

174.  Gn.  any  two  conjugate  diameters,  aa',  /3/3',  of  an  ellipse  ; 
rq.  its  principal  axes. 

175.  Gn.  the  convex  plane  pentagon  abode;  rq.  the  ellipse 
determined 

a)  by  the  five  vertices  as  pts., 

b)  by  the  five  sides  as  tangents. 

Find  the  principal  axes,  AA',  BB1,  of  the  ellipse. 

176.  Gn.  the  plane  pentagon  abode,  in  which  the  pt.  e  falls 
within  the   convex   quadrilateral  described  with  a,  b,  c,  d,  as 
vertices  ;  rq.  the  hyperbola  determined 

a)  by  the  five  pts.  as  vertices, 

b)  by  the  five  sides  as  tangents ;  the  intersections  of 

any  sixth  tangent,  pq,  with  ab  and  ac,  are  to  be 
determined. 

177.  Gn.  a  A,  abc,  and  a  direction,  mra,  in  its  plane  ;  rq.  the 
parabola  with  axis  ||  to  ran,  passing  through   a,  6,  c ;  also  the 
principal  axis,  ox,  and  the  tangent  at  its  vertex,  oy. 

178.  Gn.  any  four  pts.,  a,  b,  c,  d,  in  a  plane  and  a  tangent 
line,  pq,  passing  through  a ;  rq.  the  conic  section  thereby  de- 
termined. 

179.  Gn.  an  oblique  cylindrical  surface  with  circular  base  in 
IT  and  the  F-pr.,  a",  of  the  pt.  a  upon  it ;  rq.  a'. 

180.  Gn.  an  oblique  conical  surface  with  circular  base  in  V 
and  the  //-pr.,  a',  of  the  pt.  a  upon  it;  rq.  a". 

181.  Of  a  cylinder   of   revolution,  abc-afifr,  gn.  the  axis, 
oo,  and  the  radius,  R,  of  the  base,  abc ;  rq.  the  cylinder. 

182.  Of  a  cone  of   revolution,   S-abc,  gn.  an  element,  Sa, 
upon  which  it  rests  in  H  and  the  radius,  R,  of  the  base  abc ; 
rq.  the  cone. 

183.  Gn.   an  oblique  cone,   S-abc,  with  circular  base  in  H 
and  a  pt.,  p,  on  its  surface  ;  rq.  the  development,  (S)  —  (a)  (b) 
(c) ,  of  the  surface  and  (p)  of  the  pt. 


EXERCISES   AND   PROBLEMS.  27 

184.  Gn.   an  oblique  half-cylinder,  abc-a^c^  resting   upon 
H  with  concave    side    downwards    and    having   circles    as  the 
F-prs.  of  its  bases,  also  a  pt.,  p,  thereon  ;  rq.  the  development 
(a)  (6)  —  (a1)(6J)  in  .ffof  the  cylinder  and  (p)  of  the  pt. 

185.  Gn.  cones  of  revolution    S-abc,  with  bases  in  H  and 
planes,  t'Tt",  intersecting  them  at  various  angles  with  the  bases 
dbc ;  rq.  sections,  mno  etc.,  and  their  development  in  H,  (m) 
(n)  (o)  etc. 

186.  Gn.   an    oblique    cylinder,  dbc-a^c^    with   base  in  H 
and  a  plane,  t'Tt",  intersecting  the  cyliuder  and  _L  to  its  ele- 
ments;  rq.  the  prs.  of  the  intersection,  mno  etc.,  and  its  de- 
velopment in  H  upon  the  trace  t'T. 

187.  Gn.  an  oblique  circular  cylinder,  abc-aj)^,  with  base 
in  V  and  an  intersecting  right-line,  ab ;   rq.  the  pts.,p,  q,  in 
which  the  line  pierces  the  surface  of  the  cylinder. 

188.  Gn.  an  oblique  circular  cone,  S-abc,  with  base  in  H 
and  an  intersecting  right-line,  06 ;  rq.  p,  q,  as  in  187. 

189.  Gn.  a  sphere  with  centre,  c,  in  neither  H  nor  Fand  an 
intersecting  line,  ab  ;  rq.  p,  q,  as  in  187. 

190.  Gn.  an  oblique  circular  cone,   S-abc,   with  base  in  V 
and  a  pt.,  j?,  upon  its  surface;  rq.  the  plane,  t'Tt",  tangent  to 
S-abc  and  containing  p. 

191.  Gn.  an  oblique  circular  cylinder,  abc-afi^,  with  base  in 
H,  and  a  pt.,p,  upon  its  surface  ;  rq.  a  plane,  t'Tt",  as  in  190. 

192.  Gn.  an  oblique  circular  cone,  S-abc,  with   base  in  H 
and  a  pt.,  p,  in  space  ;  rq.  the  planes,  t' Tt",  as  in  190. 

193.  Gn.  an  oblique  circular  cylinder,  abc-a^b^  with  base 
in  Fand  a  pt.,  p,  in  space  ;  rq.  the  planes,  t'Tt",  as  in  190. 

194.  Gn.    a   cone   of    revolution,  S-abc,  with  axis   ||  to  G 
and  a  pt.,  p, 

a)  upon  its  surface, 

b)  in  space ; 

rq.  the  tangent  planes,  t'Tt",  containing  p. 

195.  Gn.  a  cylinder  of   revolution,   a&c-a^q,   with   axis  || 
to  G  and  a  direction,  mn ;  rq.  the  tangent  plane,  t'Tt",  \\  to 
the  given  direction. 


28  DESCRIPTIVE  GEOMETRY. 

NOTE.     In  exercises   196-9  and    202-7   the    cones   and  cylinders   are 
oblique,  with  circular  bases,  and  mno  etc.,  is  the  rq.  figure  of  intersection. 

196.  Gn.  a  cone,  S-abc,  with  base   in   H  and  a  direction, 
mn  ;  rq.  the  tangent  plane,  t'Tt",  \\  to  the  given  direction. 

197.  Gn.  a  cylinder,  abc-a^c^  with  base  in  V  and  a  direc- 
tion, mn ;  rq.  the  plane,  t'Tt",  as  in  196. 

198.  Gn.  a  cylinder,  K,  with  base  in  H  and  the  %.  e ;   rq. 
the  tangent  plane,  t'Tt",  making  with  .ffthe  ^  e. 

199.  Gn.  a  cone,  S-abc,  with  base  in  H  and  the  ^c;    rq. 
t'Tt",  as  in  198. 

200.  Gn.  a  developable  surface  with  a  helical  directrix  and 
a  pt.,  p,  on  its  surface  ;    rq.  the  tangent  plane,  t'Tt",  contain- 
ing p. 

201.  Gn.  a  surface  as  in  198  and  a  line,  ab,  in  space;  rq. 
the  plane,  t'Tt",  tangent  to  the  gn.  surface  and  ||  to  ab. 

202.  Gn.  two  intersecting  cylinders,  K,  L,  with  bases  in  H ; 
rq.  mno  etc. 

203.  Gn.  a  cylinder,  K,  with  base  in  V  and  an  intersecting 
prism,  P,  with  base  in  H',  rq.  mno  etc. 

204.  Gn.  a  cylinder,  K,  and  an  intersecting  prism,   S-abc, 
both  bases  in  H\  rq.  mno  etc. 

205.  Gn.  a  cylinder,  K,  with  base  in  F  and  an  intersecting 
cone,  S-abc,  with  base  in  H ;  rq.  mno  etc. 

206.  Gn.  two    intersecting    cones,   S-abc  and   X-stu,   with 
bases  in  H;  rq.  the  figure  of  intersection,  mno  etc. 

207.  Gn.  a  cone,  S-abc,  with  base  in  Fand  an  intersecting 
pyramid,  X-stuv,  with  base  in  H',  rq.  mno  etc. 

208.  Gn.  a  sphere  with  centre,  c,  not  in  H  or  V  and  a  pt.,  p, 
in  its  surface  ;  rq.  the  tangent  plane,  t'Tt",  containing  p. 

209.  Gn.  a  sphere  with  c  in  G  and  a  line,  ab ;  rq.  the  tan- 
gent planes,  t'Tt",  containing  ab. 

210.  Gn.  a  sphere  with  c  in  neither  J3"  nor   F  and  a  line, 
a&  ;  rq.  the  tangent  planes,  t'Tt",  containing  ab. 

211.  Gn.  an  hyperboloid  of  revolution  of  one    nappe  and  a 
pt.,  p,  on  its  surface  ;  rq.  the  tangent  plane,  t'Tt",  containing  p. 
Assume  the  axis  JL  to  H',  also  in  the  following  exercises  on 


EXERCISES    AND    PROBLEMS.  29 

surfaces  of  revolution.     Construct  the  figure  by  revolving  about 
the  axis  a  line  windschief  with  respect  to  it. 

212.  Gn.  an  ellipsoid  of  revolution  and  a  pt.,  p,  on  its  sur- 
face ;  rq.  the  tangent  plane,  t'Tt",  containing  p. 

213.  Gn.  any  surface  of   revolution  and  a  pt.,  p,  thereon; 
rq.  the  tangent  plane,  t'Tt",  containing  p. 

214.  Gn.  any  surface  of  revolution  and  a  line,  ab ;  rq.  the 
tangent  plane,  t'Tt",  containing  ab. 

215.  Gn.  an  ellipsoid  and  a  pt.,|>,  without  it ;  rq.  the  tangent 
cone  having  p  as  its  vertex,  also  rq.  the  curve  of  contact,  mno 
etc. 

216.  Gn.  a  sphere  and  an  intersecting  oblique  cylinder ;  rq. 
the  figure  of  intersection,  mno  etc. 

217.  Gn.  any  two  intersecting  surfaces  of   revolution  with 
axes  intersecting;  rq.  the  figure  of  intersection,  mno  etc. 

218.  Construct  the  general  hyperboloid  of  one  nappe,  whose 
right  section  is  elliptical ;  also  construct  its  asymptote  cone. 

219.  Construct  the  hyperbolic  paraboloid,  assume  a  pt.,  p, 
upon  its  surface  and  find  the  tangent  plane,  t'Tt",  containing  p. 

220.  Gn.  three  space  curves,  ab,  cd,  ef,  as  directrices  of  a 
warped   surface  and  a  pt.,  p,  in  ab  ;    rq.   the  rectilinear  ele- 
ment of  the  surface,  pq,  passing  through  p. 

221.  Gn.  two  space  curves,  ab,  cd,  as  linear  directrices  and 
the  plane,  t'Tt",  as  a  plane  directer  of  a  warped  surface;  also 
given 

a)  a  pt.,  p,  in  ab, 

b)  a  line,  mn,  in  t'Tt",  or  \\  to  it ; 

rq.  a)  pq,  the  rectilinear  element  of  the  surface  passing 

through  p, 
b)  pq,  the  rectilinear  element  of  the  surface  \\  to  mn. 

222.  Gn.  a  right-line,  ab,  _L  to  H  and  the  helix,  cd,  with  ab 
for  its  axis  ;  rq.  to  construct  upon  these  directrices  the  helicoid 
or  screw  surface,  the  rectilinear  elements  making  with  ab  the 
£  60°. 

223.  Gn.  a  helicoid  and  a  pt.,^>,  upon  its  surface;  rq.  the 
tangent  plane,  t'Tt",  passing  through  the  gn.  pt. 


30  DESCRIPTIVE    GEOMETRY. 

224.  Gn.  an  hyperboloid  of  revolution  with  axis  _L  to  H  and 
an  intersecting  plane,  t'Tt"  ;  rq.  the  figure  of  intersection  and 
its  development. 

SECTION  IV. 

Additional  Exercises  on  Lines  and  Surfaces  of  an  Order  higher 
than  the  First. 

225.  Gn.  a  pt.,p,  and  a  line,  ab ;  rq.  the  prs.  of  a  circle, 
mno,    whose   centre   lies   in   ab,    whose    circumference   passes 
through  p  and  whos'e  plane,  t'Tt",  is  J_  to  ab. 

226.  Gn.  a  plane,  t'Tt",  and  a  pt.,  S,  without  it;  rq.  the 
prs.  of  the  cone,  S-dbc,  whose  vertex  is  at  8,  whose  circular 
base  lies  in  t'Tt"  and  whose  elements  make  with  its  axis  the 
£  30°. 

227.  Gn.  in   V  a  circle  tangent  to  H  and  in  H  a  pt.,  m; 
rq.  a  pt.,  o,  so  situated  that  the  projection  of  the  circle  in  H 
from  o  as  a  centre  shall  be  an  hyperbola  with  middle  pt.  m  and 
an  asymptote  angle  of  120°. 

228.  Gn.  in  V  a  circle  tangent  to  H  and  in  Ha  line,  ab,  or 
a  pt.,  m ;  rq.  a  pt.,  o,  from  which  as  a  centre  the  circle  will  be 
projected  in  H  as  a  parabola  with  ab  as  axis  or  with  m  as 
focus. 

229.  Of  a  conic  section  gn.  two  tangents,  ab,  cd,  the  pts. 
m,  71,  as  pts.  of  contact  for  ab,  cd,  respectively  and 

a)  another  pt.,  p, 

b)  another  tangent,  qs  ; 

rq.  the  section,  its  centre  and  principal  axes. 

230.  Of  an  hyperbola  there  is  given  an  asymptote,  ab,  and 

a)  three  pts.,  c,  d,  e,  or 

b)  three  tangents,  mn,  op,  qr ; 

rq.  the  other  asymptote,  a/3,  the  principal  axes  and  the  hyper- 
bola itself. 

231.  Of  a  conic  section  there  is  known  one  of  the  foci,  F^ 
and          a)  three  pts.,  a,  b,  c,  of  the  circumference,  or 

b)  three  tangents,  mn,  op,  qr ; 
rq.  the  section,  its  centre    and  its  principal  axes. 


EXERCISES   AND    PROBLEMS.  31 

232.  A  line,  L,  of  unchangeable  length  glides  with  its  end 
pts.  in  two  intersecting  right-lines,  ab,  cd,  _L  to  each  other ; 
rq.  the  enveloped  curve  of  L,  the  path  of  any  pt.,  p,  upon  L. 
What  is  this  path  ?     Prove  your  answer. 

233.  Gn.  a  A,  abc,  of  unchangeable   form  and  in  its  plane 
two  intersecting  lines   mn,  op,  with    their  ^  =  to  the  ^  c  of 
the  A;  the  A  moves  with  its  vertices  a,  b,  in  mn,  op,  respec- 
tively ;    rq.    the  locus  of  c ;   of   the   centre,  o,  of  the  circum- 
scribed circle  ;  of  the  centre  of  gravity,  g,  of  the  A  ;  of  an}*  pt., 
x,  of  the  A. 

234.  Gn.  a  quadrilateral,  abed,  of  changing  form,  the  ver- 
tices a,  b,  are  fixed,  the  sides  be,  cd,  da,  remain  constant  in 
length;  rq.  the  curve,  mno  etc.,  enveloped  by  the  side  cd,  and 
the  locus  of  the  intersection  of  be,  da. 

235.  Gn.  a  circle,  o,  and  a  pt.,  p,  in  its  plane  ;  rq.  the  locus, 
mno  etc.,  of   the  intersections  of  all  JLs  that  can  be  drawn 
from  p  upon  the   tangents  of   the    circle  ;    construct   the   tan- 
gent at  any  pt.,  q,  and  find  the  centre  of  curvature,  p,  for  the 
same  pt. 

236.  Gn.  two  circles,  o  and  o',  in  a  plane  and  the  ^  a ;  rq. 
the  locus  of  the  intersection  of  all  pairs  of  tangents,  one  from 
each  circle,  meeting  at  the  ^  a ;  solve  when  a  =  75°  or  90°. 

237.  Gn.  a  right-line,  ab,  a  pt.,  p,  and  a  length,  L;  rq.  the 
curve,  mno  etc.,  formed  when  p  is  joined  with  every  pt.  of  ab 
and  the  length,  L,  is  measured   from  ab  upon  these  radii  vec- 
tores.      Eq.  the  curve  when  L     a)  is  greater  than,     b)  is  equal 
to,     c)  is  less  than,  the  distance  fromp  to  ab. 

238.  The  same  as  237  when  a  circle,  o,  is  given,  instead  of 
the  line  ab. 

239.  Construct  a  cycloid,  also  one  of  its  undulating  trochoids 
and  one  of  its  trochoids  with  double  pt.,   also   construct  the 
evolute  of  the  C}*cloid  and  a  tangent  at  any  pt. 

240.  Construct  an  epicycloid,  an   undulating  and  a  crossed 
epitrochoid  a)   when  the  radius  of  the  fixed   circle  is    double 
that  of  the  rolling  circle,  6)  when  the  former  equals  the  latter. 
Construct   the   evolute   of   the   epicycloid   in   case   a)    and  in 


32  DESCRIPTIVE   GEOMETRY. 

case  b)  show  that  the  epicycloid  is  identical  with  the  cardioid 
of  235. 

241.  Construct  a  hypocycloid,  an  undulating  and  a  crossed 
hypotrochoid  and  the  evolute  of  the  first  when  the  radius  of  the 
fixed  circle  is  a)  three  times,  b)   four  times,  c)  two  and  a  half 
times,  as  great  as  that  of  the  rolling  circle.     Show  that  the  hy- 
pocycloid in  case  6)  is  identical  with  the  envelope  of  a  line  of 
constant  length  moving  with  its  ends  in  two  right-lines  _L  to 
each  other. 

242.  Gn.  a  plane,  t'Tt",  and  a  pt.,  p,  in  t'T',  rq.  a  complete 
branch  of  a  cycloid  having  Tt"  for  its  base  and  tangent  to  t'T 
at  the  pt.  p. 

243.  The  axis  So  of  a  cone  of  the  2d  order  is  given  ||  to  G, 
the  planes  of  the  major  and  minor  sections  are  ||  to  F  and  H 
respectively ;  rq.  the  focus  lines   of   the    cone    and  the  three 
principal  projections  of   its  intersection  with  a  sphere  whose 
centre  is  at  its  vertex. 

244.  Gn.  two   intersecting   right-lines,  06,  cd ;    through   ab 
there  is  laid  a  complete  system  of  planes,  and  to  each  of  these 
a  J_  plane  containing  cd.     Examine  the  locus  of  the  intersec- 
tions of  the  pairs  of  J_  planes,  and  find  its  intersections  with  // 
and  F. 

245.  Upon  a  sphere  with  centre  o  lie  two  pts.,  a,  &;  two 
great  circles  always  at  right  angles  to  each  other  are  moved 
along  the  surface  of  the  sphere,  one  always  passing  through  a, 
the  other  through  b  ;  examine  the  locus  on  the  sphere  of  their 
pt.  of  intersection. 

246.  The  directrix  of  a  cone  whose  vertex  is  $,  is  a  con- 
choid, a&c,  iu  H.     This  cone  is  cut  by  a  plane,  t'Tt",  whose 
t'T  is  _L  to  the  asymptote  of  abc ;  rq.  the  projections  of  the  in- 
tersection, fofto,  and  its  true  form,  (m)(?t)(o). 

247.  Gn.  in  H  a  parabola  by  its  focus,  /,  and  principal  tan- 
gent, ab  ;  the  locus,  def,  of  the  intersections  of  J_s  from  the 
principal  vertex,  A,  with  the  tangents  of  the  parabola,  is  taken 
as  the  directrix  of  a  cone  with  given  vertex,  8 ;  rq.  the  inter- 
section, rano,  of  the  cone  with  any  plane,  t' Tt". 


EXERCISES   AND   PROBLEMS.  33 

248.  Gn.  an  oblique  cylinder  with  its  elements  ||  to  a  gn. 
line,  a&,  and  its  directrix  a  common  cycloid  in  H',  rq.  a  nor- 
mal section,  mno,  the  true  form  of  the  latter,  and  for  any  pt., 
j),  of  mno  the  tangent,  pq,  and  centre  of  curvature,  p. 

249.  Gn.  a  right-line,  ao,  a  cylinder,  aoc-a^Cj,  and  a  cone, 
S-def,  none  intersecting  either  of  the  other  two ; 

rq.  a)  planes  t'Tt",  r'Rr",  \\  to  ab  and  tangent  to  S-def, 

and   their    intersections,    mno,    m^o^   with  the 
cylinder, 

6)  planes  t'Tt",  r'Rr",  \\  to  ab  and  tangent  to  abc- 
aibici->  aQd  their  intersections,  mno,  m^o^  with 
the  cone. 

250.  Gn.  a  cone  of  revolution  with  the  diameter  of  the  base 
equal  to  one-half  the  slant  height,  and  a  pt.,  a,  upon  its  sur- 
face ;  rq.  the  curve,  a&cct,  of  shortest  distance  from  a  around 
upon  the  surface  of  the  cone  back  to  the  pt.  a  again,  and  the 
tangent  at  any  pt.,  b. 

251.  Gn.   two   circular  cylinders    (7,    Ci,   with  bases  equal 
circles  in  H  and  with  elements  so  drawn  that  a  plane  ||  to  both 
S3*stems  has  its  If  trace  ||  to  the  line  joining  the  centres  of  the 
bases ;  rq.  the  intersection,  mno,  of  the  surfaces  ;  project  iso- 
metrically  the  solid  thus  cut  out. 

252.  Gn.  a  cylinder  of  revolution,  (7,  and  a  cone  of  revolu- 
tion, R,  with  bases  in  ZT,  with  an  element  of  C  for  axis  of  J?, 
and  with  the  plane  of  the  axes  of  C  and  It  \\  to  F;  rq.  curve 
of  intersection,  mwo,  and  proof  that  its  F-pr.  is  a  parabola. 

253.  Gn.  two  cylinders  of  revolution  of  equal  normal  sec- 
tion which  pierce  each  other  at  right  angles ;  the  axis  of  each 
cylinder  is  tangent  to  the  cylindrical  surface  of  the  other ;  rq. 
the  figure  of  intersection,  mno,  its   isometric   projection,   the 
development  of  one  of  the  cylinders,  and  the  tangent  at  the  pt. 
p  of  the  developed  curve. 

254.  Gn.  two  windschief  lines,  rco,  ccZ,  and  in  ab  the  pt.  p  ;  rq. 
the  prs.  of  a  helix  which  has  ao,  cd,  for  principal  normals  and 
passes  through  the   pt.  p.     That  is,  ao,  cd,  cut  the  helix  at 
right  angles  and  are  tangent  to  the  cylinder  of  revolution  upon 
which  the  helix  is  wound  ;  rq.  two  spires  of  the  helix. 


34  DESCRIPTIVE    GEOMETRY. 

255.  Gn.  a  developable  helical  surface  ; 

rq.  a)   its   intersection,   mno   etc.,  with  a  plane  passing 

through  the  axis.     Assume  axis  J_  to  H  and  cut 
by  a  plane  _L  to  V\ 

b)  its  intersection,  rst  etc.,  with  any  plane, 

c)  its  development  in  H. 

256.  Gn.   a  developable   or   tangential   helical   surface  and 
upon  it  two  pts.,  a,  6;  rq.  the  shortest  path  along  the  surface 
from  a  to  b. 

257.  Gn.  two  cones  of  revolution,   (7,  C",  with  elements  the 
same  length,  but  with  the  diameter  of  the  base  of  C'  one-half 
that  of  C ;  their  vertices  are  made  to  coincide  and  C'  is  rolled 
externally  upon  C,  the  two  surfaces  always  having  one  and  only 
one  element  in  common  ;  rq.  one  complete  branch,  mno,  of  the 
locus  of  any  pt.,  a,  in  the   circumference  of  the   base  of  C", 
also  corresponding  conical  crossed  and  inflected  epitrochoids. 

258.  Gn.  two  windschief  lines,  ab,  cd ;  rq.  two  single  sur- 
faced hyperboloids  of  revolution  having  ab  and  cd  for  axes  and 
designed  to  work  together  tangentially,  transmitting  motions 

a)  equal, 

b)  in  which   the   angular   velocity  of   hyperboloid  ab 
is  to  that  of  hyperboloid  cd  as  2:3. 

259.  Gn.  in  space  three  windschief  lines,  a&,  cd,  e/;  rq.  the 
hyperboloid  of  one  nappe  constructed  upon  these  lines  as  direc- 
trices, its  axis  and  the  ellipse  of  its  gorge. 

260.  Gn.  two  parabolas  with  the  directions  from  their  prin- 
cipal  vertices   to   their   foci  exactly  opposite    and    with  their 
planes  always  at  right  angles.     One  parabola  moves  as  a  gen- 
eratrix with  its  principal  vertex  gliding  in  the  other  parabola. 
All  positions  of  the  two  axes  are  ||   and  all  positions  of  the 
plane    of    the    generating    parabola    are    ||  ;    rq.    the   surface 
described. 

261.  Gn.  a  cone  of  revolution  and  a  co-axial  helicoid  ;  rq. 
the  intersection,  mno  etc.,  of  the  two  surfaces. 

262.  Gn.   two  windschief   lines,  «6,  cd,  and  a  line,  L,  of 
limited  but  fixed  length  longer  than  the  common  _L  of  a&,  cd. 


EXERCISES   AND   PROBLEMS.  35 

L  moves  with  one  end  in  afr,  the  other  in  cd ;  rq.  the  surface 
described  and  its  intersection,  mno  etc.,  by  a  plane,  t'Tt",  _L 
to  the  common  _L  of  a&,  cc?,  at  its  middle  pt.,  p,  and  the  devel- 
opment of  mno  etc.,  upon  t' T. 

263.  Gn.  as  meridian  section  of  a  surface  of  revolution  two 
equal  circles,  the  distance  of   whose  centres  is  two-thirds  the 
diameter  of  either  circle,  the  axis  of  the  surface  being  the  com- 
mon chord.     Intersect  this  surface  by  a  plane  ||  to  its  axis  and 
bisecting  the  radius  of  its  equator;  rq.  the  section,  mno  etc., 
which  is  a  Cassinian  curve. 

264.  Gn.   a  complete  spire   of  a  helix  and   a  sphere  with 
radius  one-fourth  of  that  of  the  cylinder  upon  which  the  helix 
is  wound ;  rq.  the  surface  described  when  the  sphere  moves 
with  its  centre  in  the  helix. 

265.  Gn.  a  staircase   vestibule  12  feet  square  and   14  feet 
high.  A  spiral  staircase  begins  at  the  middle  pt.  of  one  side  and 
ends  on  the  floor  above,  after  having  wound  through  an  arc  of 
270°.     It  leaves  an  opening  in  the  form  of  a  cylinder  of  revolu- 
tion with  axis  vertical,  whose  shortest  distance  from  any  one  of 
the  three  sides  bearing  the  staircase  is  3  feet.     Each  stair  is  6 
inches  high,  and  a  balustrade  two  feet  high,  starting  from  a 
newel  post  at  the  bottom,  winds  about  the  well-hole  to  the  top  ; 
rq.  the  projections  of  the  staircase.     Design  the  balustrade,  its 
supports  and  the  newel  post  according  to  taste. 


PART    II. 

SUGGESTIONS,  ANALYSES  AND  DEMONSTRATIONS. 

SECTION  I. 
Point,  Line,  Plane. 

PROBLEM  1.  Figure  1  of  the  plates  shows  in  isometric  per- 
spective the  pt.  a  in  space  in  the  first  quadrant  and  its  pro- 
jections; Fig.  2  shows  the  orthographic  transformation  of  the 
same.  In  general,  a  pt.  will  be  found  in  some  one  of  the  four 
quadrants.  The  student  will  easily  see  the  transformations 
to  be  made  when  the  pt.  lies  in  the  second,  third,  or  fourth 
quadrant.  Special  positions  of  the  pt.  will  arise  when  it  lies  in 
some  projecting  plane,  or  ground-line,  or  plane  bisecting  the 
diedral  angle  between  two  planes  of  projection. 

PROB.  3.  Fig.  3  shows  in  isometric  perspective  the  line  ab 
crossing  the  first  quadrant,  while  Fig.  4  shows  the  same  line  in 
projection  ;  it  will  be  noted  that  the  H-pr.  above  G,  or  to  the 
right  of  Cr2,  is  regarded  as  covered  by  F,  respectively  P,  and 
is  represented  therefore  by  short  dashes  ;  similarly  the  F-pr. 
below  Cr,  or  to  the  right  of  6r2,  and  the  P-pr.  to  the  left  of  G2, 
or  below  O ;  for  G3  regarded  as  belonging  to  P  coincides  after 
transformation  with  G.  The  student  should  be  careful  to 
designate  in  each  drawing  the  pts.  h,  v,  p.  In  the  line  ab,  li  is 
the  pt.  of  meeting  of  H,  the  H  projecting  plane  of  ab  and  its 
V  projecting  plane.  The  last  and  H  form  two  intersecting 
planes,  both  _L  to  a  third  or  V.  Their  line  of  intersection 
passing  through  h  is  therefore  _L  to  F,  therefore  to  G  at  the  pt. 
where  the  F-pr.  meets  G.  Therefore  to  find  h  we  have  the 
directions  given  in  Art  15,  of  Part  III.  Similarly  for  v  and^?. 
Special  positions  of  the  line  occur  when  it  is  H  to  one  or  more 


SUGGESTIONS,  ANALYSES,  AND   DEMONSTRATIONS.      37 

of  the  planes  of  projection,  or  lies  in  one  or  two  of  those  planes, 
or  in  one  of  the  bisecting  planes  of  the  diedral  angles  formed 
by  an}'  two  of  the  fundamental  planes.  Rigorously  taken  there 
are  eight  angular  spaces  bounded  by  the  three  fundamental 
planes,  but  we  shall  understand  by  the  four  quadrants  the  four 
diedral  angles  formed  by  the  intersection  of  H  and  V. 

PROB.  4.  From  this  point  on,  the  third  projection  may  be 
omitted  except  in  the  special  cases  which  require  its  use. 

PROB.  7.  Lines  lying  in  a  plane  ||  to  P  are  not  fully  known 
unless  their  P-pr.  is  given  or  their  v  and  li. 

PROB.  9.  The  line  joining  the  H-  and  F-prs.  of  a  pt.  must 
always  be  _L  to  G. 

PROB.  11.  Fig.  7  represents  a  cone  of  revolution  with  base 
in  H  and  axis  in  F;  a'  is  the  inclination  of  the  elements  to 
H.  Fig.  8  represents  a  cylinder  of  revolution  with  base  in  H 
and  the  axis  not  in  V.  Fig.  9  represents  a  sphere  with  centre 
in  G.  Fig.  10,  a  sphere  with  centre  in  space  in  the  first  quad- 
rant. Simple  changes  of  these  figures  in  position  will  solve 
problems  11  and  12. 

NOTE.  The  student  will  find  that  the  ease  with  which  a  problem  is 
solved  depends  greatly  upon  the  skill  with  which  the  data  is  assumed. 
Study  the  simplest  representations  consistent  with  generality.  Work  also 
in  the  various  quadrants  and  with  various  positions. 

PROB.  13.  Take  a'b'  between  h  and  G  as  the  base  of  a  right 
2£d  A,  and  the  distance  from  v  to  G  as  the  _L.  That  is,  from 
the  pt.  where  a'b1  cuts  G  draw  in  H  an  indefinite  J_  to  a'b'.  With 
the  same  pt.  in  G  as  a  centre  and  its  distance  to  v  as  a  radius, 
describe  a  circle  till  it  intersects  the  _L  drawn  to  a'b'  in  H. 
This  pt.  of  intersection  joined  to  h  gives  L  and  its  inclination 
to  a'b'  gives  a'.  Proceed  in  the  same  wa}T  with  a"b". 

PROB.  14.  Since  the  line  connecting  two  points  in  space  is 
the  hypotenuse  of  a  right  ^d  A,  of  which  the  distance  between 
the  If  or  F-prs.  is  the  base  and  the  difference  between  the  V 
or  H  projecting  lines  the  _L,  it  is  sufficient  to  construct  a  right 
^d  A  with  these  legs.  Or  one  pr.  of  the  connecting  line  ab 
can  be  revolved  about  one  of  its  extremities  till  ||  with  #,  when 


38  DESCRIPTIVE   GEOMETRY. 

the  line  in  its  new  position  will  be  ||  to  the  other  plane  and  its 
new  projection  in  that  plane  will  be  its  true  length. 

PROB.  17.    One  solution  is  shown  in  Fig.  11. 

PROB.  19.  The  %.  between  the  JT-pr.  and  G  is  the  pr.  of  a". 
Construct  an  %.  equal  to  a"  and  ||  to  H,  one  side  being  in  V 
and  vertex  at  v.  Revolve  this  2£  about  a  _L  to  V  through  v  as 
an  axis,  until  its  H-pr.  as  shown  by  some  pt.  in  the  line  coin- 
ciding with  a  pt.  of  the  side  of  the  2£,  coincides  with  the  given 
H-pr.  The  F"-pr.  belonging  to  this  position  fully  determines 
the  line. 

PROB.  20.  Construct  a  rectangular  parallelepiped  with  one 
face  in  H  and  one  in  F,  such  that  the  diagonal  makes  the  ^  a' 
with  the  diagonal  of  the  face  in  H  and  the  ^  a"  with  the  diag- 
onal of  the  face  in  F.  Pass  a  ||  line  through  the  pt.  h. 

PROB.  21.  Solve  by  first  finding  p'",  using  as  an  auxiliary 
line  the  bisector  of  the  ^  between  G2  and  G3. 

PROB.  22.  The  representation  of  a  plane  in  space  and  in 
projection  is  shown  in  Figs.  5  and  6.  It  will  be  observed  that 
when  a  line  lies  in  a  plane  its  traces,  /*,  w,  ^9,  must  always  be 
found  in  the  corresponding  traces  of  the  plane. 

PROB.  23.  Use  as  an  auxiliary  figure  any  line  of  the  plane 
containing  the  required  pt. 

PROB.  24.  Use  as  auxiliary  figures  planes  _L  to  t'T  and  Tt". 
Revolve  the  intersections  upon  the  H  and  V  traces  of  these 
planes  into  H  and  F  respectively.  Fig.  6  shows  how  e'  may 
be  found.  A  similar  construction  should  be  used  for  e". 

PROB.  27.  Construct  a  right  cone  of  revolution  with  axis  in 
F-L  to  H  and  elements  making  an  ^  e'  with  H\  a  plane  tan- 
gent to  this  cone  makes  the  required  %.  with  H.  Upon  the 
elements  lying  in  F  construct  a  right  A  right-angled  at  G 
and  with  opposite  acute  ^  =  to  K',  the  hypotenuse  will  be  the 
length  of  the  F  trace  between  the  vertex  of  the  cone  and  G, 
while  the  H  trace  will  be  tangent  to  the  circular  base  of  the 
cone. 

PROB.  28.  Find  the  vertex  of  a  cone  of  revolution  whoso 
axis  is  in  F,  base  in  JET,  inclination  of  elements  to  H,  e',  and  to 


SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS.      39 

whose  base  tT'  is  tangent.  Tt"  connects  the  vertex  of  this 
cone  with  the  pt.  in  which  t'T  intersects  G. 

PROB.  29.  The  auxiliary  cone  now  has  its  axis  in  If  and 
base  in  F. 

PROB.  30.  If  a  sphere  be  taken  with  centre  in  G  and  two 
cones  of  revolution  tangent  to  this  sphere,  one  with  vertex  in 
F,  axis  _L  to  H  and  elements  making  ^  c'  with  H,  the  other 
with  vertex  in  H*  axis  _L  to  V  and  elements  making  ^  e"  with 
F,  a  plane  tangent  to  both  cones,  i.e.  whose  traces  pass  through 
the  vertices  of  the  cones  and  are  tangent  to  their  bases  will  be 
II  to  the  plane  sought.  The  construction  is  given  in  Fig.  12. 

PROB.  31.  The  revolution  upon  t'T  rq.  is  shown  in  Fig.  6. 
Let  the  student  solve  the  problem  when  the  pt.  alone  is  devel- 
oped. He  should  also  solve  the  problem  when  K  is  greater 
than  90°. 

PROB.  33.  Revolve  into  H  upon  the  line  (p)  p'  as  an  axis 
the  rt.  A  p'  (p)  p".  It  will  be  seen  that  the  basal  £  of  this  rt. 
A  =  ^e'.  Therefore  bisect  its  hypotenuse  by  a  _L  ;  the  rq.  t'T 
must  pass  through  the  intersection  of  the  J_  with  the  base  of 
the  A  and  must  be  perpendicular  to  that  base. 

PROB.  35.  The  direction  of  the  H  trace  is  known,  and  since 
with  the  gn.  2£  K  we  can  construct  an  auxiliary  plane  ||  to  the 
rq.  one  with  its  IT  trace  passing  through  the  developed  pt.,  e'  is 
known  and  the  problem  reduces  to  29. 

PROB.  36.  When  the  £  K  is  developed  into  H  or  F,  the  loci 
of  the  developed  position  of  the  rq.  pt.  will  be  lines  ||  to  the  If 
and  F  traces  at  the  distances  m',  m",  respectively.  The  inter- 
section of  these  loci  will  be  the  developed  pt. 

PROB.  38.  The  line  joining  the  H  traces  or  F  traces  of 
the  lines  will  be  the  H  or  F  trace  respectively  of  the  plane. 
The  ^  8  is  found  by  developing  the  plane  and  lines  in  H 
or  F. 

The  data  necessary  for  determining  a  plane  are  given  in  Art. 
19  of  Part.  III. 

PROB.  39.  Develop  the  given  plane  and  line,  make  the  rq. 
construction  and  find  the  rq.  line  by  counter-revolution. 


10  DESCRIPTIVE   GEOMETRY. 

PROB.  40.  The  first  part  is  solved  the  same  as  the  first  part 
of  38  ;  the  latter  part  by  development. 

The  student  should  prove  that  the  projections  of  two  parallel 
lines  are  respectively  parallel  lines. 

PROB.  41.    Fig.  13  gives  the  solution. 

PROB.  42.  The  rq.  plane  is  found  by  connecting  the  gn.  pt. 
with  any  pt.  of  the  gn.  line  and  thus  reducing  that  part  of  the 
problem  to  38. 

PROB.  44.  The  gn.  pt.  is  one  pt.  in  the  rq.  line.  With  the 
distance  of  the  F-pr.  of  the  pt.  from  G  as  a  _L  and  8  as  a  basal 
angle  form  a  rt.  A.  With  the  base  of  this  as  a  radius  and 
the  H-pr.  of  the  pt.  as  a  centre,  describe  the  base  of  a  cone  of 
revolution.  Its  intersection  with  the  H  trace  of  the  gn.  plane 
is  a  second  pt.  in  the  rq.  line. 

PROB.  45.  t'Tmusi  pass  through  h  and  must  be  tangent  to 
the  base  of  a  cone  of  revolution  whose  axis  is  the  _L  from  v  to 
G  and  one  of  whose  elements  is  the  JL  from  v  upon  t'T. 

This  element  is  found  by  revolving  ab  upon  a'b'  into  H,  con- 
structing through  h  a  line  making  the  %.  8  with  a^  and  taking 
the  perpendicular  from  the  revolved  position  of  v  to  this  line. 
Knowing  the  axis  and  element  of  the  auxiliary  cone,  the  radius 
of  the  base  is  easily  found.  With  this  radius  and  with  the 
centre  at  the  foot  of  the  _L  from  v  to  G,  a  circle  is  constructed. 
The  rq.  H  trace  passes  through  li  and  is  tangent  to  this  circle. 
The  V  trace  passes  through  v. 

PROB.  48.  Construct  the  line  cut  out  from  the  gn.  plane  by 
the  Hor  F  projecting  planes  of  the  line.  The  intersection  of  the 
F-pr.  or  -£T-pr.  of  this  line  with  the  F-pr.  or  JET-pr.  of  the  given 
line  must  indicate  an  actual  intersection  in  space  and  be  one 
pr.  of  the  pt.  sought.  When  the  gn.  line  lies  in  a  plane  _L  to 
G,  its  h  and  v  must  be  given  and  the  problem  is  solved  by 
using  this  _L  plane  as  a  third  plane  of  projection.  Fig.  14  gives 
the  first  of  the  four  rq.  cases  when  K  is  obtuse.  m"w"  is 
the  F-pr.  of  the  line  of  t'Tt"  cut  out  by  the  H  projecting 
plane  of  ab.  q'r'  is  the  JJ-pr.  of  the  line  of  t'Tt"  cut  out  by 
the  F  projecting  plane  of  ab. 


SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS.      41 

PROB.  50.  Since  the  gn.  line  is  _L  to  the  rq.  plane,  the 
directions  of  the  rq.  traces  are  known  by  Part  III.,  Art.  24. 
Through  the  gn.  pt.  p  draw  a  line  ||  to  the  .ff  or  V  trace  of  the 
rq.  plane.  This  auxiliary  line  must  be  a  line  of  the  rq.  plane 
and  the  pt.  where  it  pierces  the  V  or  H  plane  must  be  a  pt.  in 
the  V  or  H  trace  of  the  rq.  plane.  One  pt.  in  either  trace 
being  known,  the  plane  is  known. 

PROB.  51.  The  li  or  v  traces  of  the  gn.  line  and  the  con- 
structed J_  determine  the  H  or  V  trace  of  a  plane  ;  this  trace 
may  be  used  as  an  axis  upon  which  to  revolve  the  angle  of  the 
lines  ab  and  ph  into  H  or  F 

PROB.  53.  The  rq.  line  must  be  an  element  of  the  cone  of 
revolution  whose  vertex  is  at  the  gn.  pt.,  whose  axis  is  _L  to 
H  and  whose  elements  make  the  £  8  with  H.  Construct  such 
a  cone.  Then  to  determine  which  element,  pass  a  line  through 
the  vertex  of  the  cone  _L  to  the  gn.  plane.  Find  the  pt.  in 
which  it  pierces  H.  This  _L  must  make  with  the  rq.  element 
an  ^  =  90°—  0.  Construct  a  A,  one  of  whose  sides  shall  be  the 
true  length  of  the  J_  to  t'Tt"  from  p  to  JJ;  the  second,  the  true 
length  of  an  element  of  the  cone  ;  the  included  %  =  90°  — 0.  The 
third  side  is  the  distance  from  the  foot  of  the  J_  to  the  foot  of 
a  rq.  element.  How  many  solutions  ? 

PROB.  54.  When  the  //  and  V  traces  respectively  intersect 
within  the  limits  of  the  paper,  their  pts.  of  intersection  are  the  h 
and  v  of  the  rq.  line  of  intersection.  When  the  H  and  V 
traces  do  not  so  intersect,  pass  auxiliary  planes  II  to  V  and  at 
such  a  distance  that  the  F-prs.  of  the  lines  cut  from  the  gn. 
planes  do  intersect  within  the  necessary  limits.  Two  of  these 
intersections  will  determine  the  F-pr.  of  the  line  of  intersec- 
tion. A  similar  construction  will  determine  its  JET-pr.  Other 
auxiliary  constructions  will  occur  to  the  thoughtful  student. 

PROB.  55.  Draw  the  JET  trace  of  an  auxiliary  plane  J_  to  the 
line  of  intersection  of  the  two  planes.  In  order  to  determine  by 
revolution  upon  this  H  trace  the  plane  angle  of  the  two  planes, 
it  only  remains  to  find  where  the  line  of  intersection  pierces 
the  auxiliary  plane. 


42  DESCRIPTIVE    GEOMETRY. 

Revolve  the  line  of  intersection  upon  its  H-pr.  into  H.  A  _L 
let  fall  upon  the  revolved  line  of  intersection  from  the  pt.  of 
intersection  of  the  auxiliar}-  H  trace  with  the  .fl-pr.  of  the  line 
of  intersection,  will  determine  the  required  pt.  Revolve  the 
auxiliary  plane  upon  its  H  trace  into  //.  The  developed  pt.  of 
piercing  joined  with  the  pts.  in  which  the  auxiliary  H  trace 
crosses  the  H  traces  of  the  given  planes,  will  form  the  angle 
sought.  A  similar  construction  may  be  made  in  F. 

PROB.  56.  Assume  a  third  plane  _L  to  II  or  Fand  the  H  or 
F  traces  of  the  gn.  planes.  Revolve  this  auxiliary  plane  upon 
its  Hor  F  trace  into  H  or  F.  The  distance  of  the  Us  cut  out 
is  the  required  distance. 

PROB.  57.  Solved  by  an  auxiliary  line  through  a  pt.  in  ab 
and  II  to  cd,  thus  giving  two  intersecting  lines  to  determine  the 
rq.  plane. 

PROB.  58.  Two  auxiliary  lines  are  passed  through  the  gn. 
pt.,  p,  II  to  the  gn.  lines,  ab  and  cd. 

PROB.  59.  If  through  ab  a  plane  be  passed  II  to  cd,  by  53, 
and  cd  be  projected  upon  this  plane,  it  is  evident  that  the  pt. 
in  which  this  pr.  cuts  ab  is  the  foot  of  the  common  _L  of  the 
two  lines. 

Therefore,  as  in  57,  construct  a  plane  containing  ab  and  II  to 
cd.  From  any  pt.  of  cd  draw  a  J_  to  the  plane  and  find  its 
foot.  The  line  through  this  foot  II  to  cd  is  the  pr.  of  the  latter 
upon  the  plane  and  the  pt.  where  this  pr.  intersects  ab  is  the 
foot  of  the  rq.  common  _L.  The  latter  is  found  by  drawing 
from  its  foot  a  _L  to  the  plane  and  limiting  this  _L  by  the  plane 
and  cd. 

PROB.  60.  a)  If  through  the  gn.  pt.  and  each  of  the  gn. 
lines  planes  be  passed,  their  intersection  will  be  the  rq.  line. 

b)  Through  p  and  any  pt.  of  ab  pass  an  auxiliary  line  inn. 
Find  the  pts.  where  ab  and  mn  pierce  the  .BT  projecting  plane 
of  cd,  call  these  pts.  #,  y,  respectively.  The  line  xy  must  be 
the  intersection  of  the  plane  of  ab  and  mn  with  the  //  project- 
ing plane  of  cd.  The  intersection  of  x"y"  with  c"d"  must  be 
the  F-pr.  of  the  pt.  q,  where  cd  pierces  the  plane  of  a&,  mn. 
Join  p  and  q.  Find  a  proof  of  your  work. 


SUGGESTIONS,   ANALYSES,   AND  DEMONSTRATIONS.     43 

PKOB.  63.  Draw  through  p  a  line  _L  to  t'Tt"  and  another 
line  ||  to  the  given  line.  These  two  intersecting  lines  will 
determine  r'Er". 

PROB.  64.  Consider  the  gn.  line  the  line  of  intersection  be- 
tween the  gn.  and  rq.  planes  and  reverse  the  process  of  55. 

PROB.  65.  If  any  pt.  of  the  gn.  line  be  chosen  except  its 
foot  in  the  gn.  plan'e  and  this  pt.  be  taken  as  the  vertex  of  a 
cone  of  revolution  whose  axis  is  _1_  to  the  gn.  plane  and  whose 
elements  make  with  that  plane  the  2£<£,  it  is  evident  that  the 
tangent  to  the  base  of  this  auxiliary  cone  from  the  pt.  where 
the  gn.  line  pierces  the  gn.  plane,  will  be  a  second  line  for  the 
determination  of  the  rq.  plane.  Therefore,  find  the  pt.  where 
the  gn.  line  pierces  the  gn.  plane.  Choose  another  pt.  of  the 
line  and  find  where  the  J_  from  this  pt.  to  the  gn.  plane  pierces 
the  latter  and  the  length  of  that  J_.  The  latter  with  the  ^  </>  gives 
the  radius  of  the  base  of  the  auxiliary  cone  of  revolution.  De- 
velop the  plane  with  the  pts.  found  into  H  or  F.  Construct 
the  base  of  the  cone  and  the  rq.  tangent.  By  counter-revolu- 
tion the  two  necessaiy  intersecting  lines  will  appear. 

PROB.  66.  Consider  the  gn.  pt.  the  vertex  of  a  cone  of  rev- 
olution whose  axis  is  _L  to  -the  gn.  plane  and  whose  elements 
make  the  2£  8  with  that  plane.  Find  the  axis  of  this  cone,  the 
radius  of  its  base  and  the  pt.  where  its  axis  pierces  the  gn. 
plane.  In  the  developed  position  of  the  gn.  plane  find  the  pt. 
in  which  the  base  of  the  cone  cuts  the  gn.  line.  By  counter- 
revolution two  pts.  will  appear  for  determining  the  rq.  line. 
In  general,  two  solutions. 

PROB.  67.  Designate  the  face  2£s  by  a,  /?,  y,  their  opposite 
dihedral  ^  s  by  A,  B,  C.  Assume  one  face  £,  as  /?,  in  H  with 
the  edge  between  a  and  ft  _L  to  G.  Construct  the  ^  a  in  H 
adjacent  to  and  on  one  side  of  ft ;  the  %.  y  in  H  adjacent  to 
and  on  the  other  side  of  (3.  Let  the  common  vertex  of  a,  (3, 
y,  be  0.  This  will  be  the  vertex  of  two  cones  of  revolution, 
any  pt.  besides  (0)  in  whose  common  element  is  sought.  To 
obtain  a  second  pt.  we  determine  where  this  common  element 
-  the  edge  between  a  and  y  -  pierces  V.  The  locus  of  this 


44  DESCRIPTIVE   GEOMETRY. 

pt.  in  F  on  the  side  of  a  is  a  circle,  the  base  of  a  cone  of 
revolution,  whose  axis  is  on  one  side  of  the  *%.  a  and  generat- 
ing element  the  other  side.  If  the  length  of  this  generating 
element  be  laid  off  on  the  outer  side  of  the  ^  y,  the  distance 
from  its  extremity  to  the  intersection  in  G  of  the  edge  between 
/3  and  y  will  be  the  distance  from  the  latter  pt.  to  the  rq.  pt. 
in  V.  The  £  C  will  be  directly  gn.,  while  the  %  s  A  and  B 
are  found  by  methods  already  explained. 

This  problem  is  constructed  in  Fig.  15. 

PROB.  68.  Let  a,  /?,  (7,  be  given  in  positions  described  in 
67  ;  the  trihedral  is  easily  found  by  a  slight  modification  of  the 
same  problem. 

PROB.  69.  Let  a,  /?,  A,  be  the  three  parts  given  as  above. 
By  reversing  the  operation  of  finding  A  in  68  the  F  trace  of 
the  face  2£  y  will  be  known,  and  if  the  data  are  so  taken  that 
the  problem  is  possible,  the  pts.  (in  general  two)  of  the  inter- 
section of  the  third  edge  with  F  are  known. 

PROB.  72.  Place  the  2£  A  in  position,  its  edge  being  in  //_L 
to  G.  There  will  then  be  two  planes  in  position,  one  in  H  con- 
taining the  yet  unknown  face  ^  /?,  another  _L  to  F,  making 
the  ^  A  with  H  and  containing  the  unknown  face  £  y.  Tan- 
gent to  the  latter  plane  construct  an  auxiliary  sphere  with  its 
centre  in  G.  Tangent  to  this  sphere  construct  two  cones  of 
revolution  with  axes  in  F,  the  elements  of  one  making  with  H 
the  ^  C,  the  elements  of  the  other  making  with  the  plane  of 
y  the  2£  B.  Both  vertices  of  these  cones  must  lie  in  F  and 
must  be  pts.  in  the  trace  of  the  rq.  plane.  The  F  trace  of  the 
latter  is  therefore  known,  and  also  its  H  trace,  since  this  is 
tangent  to  the  base  of  the  first  auxiliary  cone.  Find  a,  /?,  y. 

PROB.  73.  Construct  an  auxiliary  trihedral  with  a  and  (ft  -+-  y) 
adjacent  face  ^s  and  C  the  included  dihedral.  Let  (ft  +  y)  be 
taken  in  H  and  the  edge  of  the  dihedral  be  _L  to  G.  Lay  off 
from  0  upon  the  outer  side  of  (ft  +  y)  a  line  equal  to  the  outer 
side  of  a  between  0  and  G.  A  line  joining  the  pt.  thus  deter- 
mined with  the  v  of  the  upper  edge  of  the  auxiliary  trihedral 
will  form  the  base  of  an  isosceles  triangle.  A  plane  through 


SUGGESTIONS,   ANALYSES,   AND  DEMONSTRATIONS.     45 

O  J_  to   this   base   gives  the  necessary  division   of  (/?  +  y). 
Why? 

PROB.  74.    This  problem  is  an  easy  modification  of  73. 

PROB.  75.  In  finding  the  F-pr.  of  the  pentagon  it  will  be 
most  direct  to  use  as  auxiliary  lines  of  the  plane  those  whose 
H-prs.  pass  through  a',  6',  c',  d',  e'. 

To  find  the  true  figure,  take  a  line  II  to  H  through  some  con- 
venient vertex  of  the  F-pr.  Join  the  corresponding  pts.  in  H. 
This  will  be  an  axis  upon  which  the  figure  is  to  be  revolved 
until  it  is  II  to  jET,  when  it  will  be  projected  in  H  in  its  true 
magnitude. 

PROB.  76.  Find  the  F-pr.  of  the  pts.  d,  e,  by  using  the 
Axis  of  Affinity  of  the  two  prs.  of  the  pentagon.  This  axis 
may  be  defined  as  follows :  when  two  plane  figures  are  so 
related  that  the  intersections  of  homologous  sides  lie  upon 
one  right  line,  the  latter  is  called  an  Axis  of  Affinity.  The 
most  general  proof  for  its  existence  depends  upon  the  following 
theorem  of  Desargues  :  — 

"  If  each  of  two  triangles  has  one  vertex  in  each  of  three 
concurrent  lines,  then  the  intersections  of  corresponding  lines 
lie  in  a  line,  those  sides  being  called  corresponding  which  are 
opposite  to  vertices  on  the  same  line." 

This  proposition  is  conveniently  demonstrated  by  the  methods 
of  Protective  Geometry.  Such  a  demonstration1  may  be  found 
on  page  394  of  the  tenth  volume  of  "Encyclopedia  Britan- 
nica." 

In  Fig.  16  let  ABC,  A'B'C',  be  two  A  with  lines  a,  6,  c, 
through  homologous  vertices  meeting  in  S  ;  then  the  homologous 
sides  meet  in  some  right  line  as  S^  £2>  $3-  This  will  still  be 
true  when  S  retreats  to  infinity  and  a,  6,  c,  become  parallel 
lines,  as  always  occurs  in  any  two  orthographic  prs.  of  a  plane 
figure.  It  also  follows  that  for  any  pr.  of  a  plane  figure  and 
the  corresponding  development  of  the  figure,  the  axis  of  de- 
velopment must  be  an  axis  of  affinity. 

Prob.  76  is  solved  in  Fig.  17. 

1  See  also  Chauvenet's  Geom.  p.  342. 


46  DESCRIPTIVE   GEOMETRY. 

PROB.  77.  Let  the  side  of  known  length  be  the  side  a&, 
having  an  extremity  in  the  vertex  a.  The  H-pr.  and  true 
length  of  this  side  being  known,  its  F-pr.  is  known  and  the  2£ 
it  makes  with  H.  Pass  an  auxiliary  plane  through  a  J_  to  ab. 
The  inclination  of  this  plane  to  H  is  the  complement  of  the 
inclination  to  H  of  ab.  The  other  side,  ad1,  having  an  extrem- 
ity in  the  vertex  a,  must  lie  in  this  plane.  By  developing  the 
latter  on  its  H  trace  the  true  length  of  ad  is  known  and  thus 
the  whole  figure. 

PROB.  78.  Construct  the  surface  right  line  passing  through 
the  vertex  and  the  F-pr.  of  the  pt.  The  H-pr.  must  lie  in  this 
line.  The  base  of  the  pyramid  is  assumed  in  H.  The  con- 
struction is  carried  out  in  Fig.  18. 

PROB.  79.  With  the  vertex  as  a  centre  revolve  the  F-pr.  of 
the  lines  forming  the  edges  until  they  are  II  to  H,  when  the 
H-pi'B.  will  be  the  true  length  and  will  show  the  inclinations. 
In  the  development  place  the  triangles  forming  the  lateral  sur- 
face adjacent  to  each  other  in  F.  The  inclinations  of  the  faces 
to  each  other  are  found  by  55. 

PROB.  80.  Construct  the  lateral  faces  in  H,  each  upon  its 
basal  side.  The  vertex  of  the  tetrahedron  is  found  by  revolv- 
ing any  two  of  the  developed  lateral  faces  upon  their  bases 
until  their  vertices  unite. 

PROB.  81.  Take  the  base  in  H  and  construct  adjacent  to  this 
in  its  proper  position  the  developed  face  safe,  whose  edges  are 
known.  The  H-pic.  of  the  vertex  s  must  lie  in  the  _L,  or  the 
JL  produced,  drawn  from  the  vertex  of  this  face  to  its  base, 
ab.  The  _L  itself  is  the  hypotenuse  and  the  gn.  altitude  the  _L 
of  a  rt.  A  whose  base  is  the  distance  from  ab  to  the  H-pv. 
of  s. 

PROB.  82.  Place  the  three  face  2£s  a,  /?,  y,  adjacent  in  H. 
The  bases  of  the  triangular  faces  in  the  developed  position  must 
all  be  tangent  to  the  circle  whose  centre  is  at  the  developed 
vertex  and  whose  radius  is  the  _L  distance  from  this  centre  to 
any  basal  edge.  Therefore  each  face  ^  must  be  divided  into 
two  segments,  each  of  which  must  be  equal  to  the  divisions 


SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS.      47 

lying  adjacent  in  the  adjacent  face  £.      Let  the  divisions  of  a 
be  x  and  y.     These  two  equations  must  then  be  true  : 

a,  fl-X  =  y-9.     .•  .  X  =  «  +     =*,    y  =  <^ 


The  known  base  is  then  introduced  _L  to  the  dividing  line  of 
its  face.  The  construction  for  the  remaining  two  bases  easily 
follows.  In  dividing  a  care  must  be  taken  to  place  the  greater 
of  the  two  divisions  adjacent  to  the  greater  of  the  two  remain- 
ing face  2£s. 

PROB.  83.  The  dihedrals  can  be  found  by  Prob.  67.  If  the 
known  lateral  edges  and  inclination  belong  to  the  same  face,  the 
base  is  directly  introduced.  If  not,  the  problem  is  reduced  to  65. 

PROB.  84.  Take  the  base  in  H.  The  centre  of  the  circum- 
scribed sphere  must  lie  in  the  _L  erected  to  the  base  at  its 
middle  pt.  It  must  .also  lie  in  a  plane  constructed  _L  to  a  lat- 
eral edge  at  its  middle  pt.  It  must,  therefore,  be  the  pt.  where 
the  J_  pierces  the  plane.  The  radius  is  the  distance  from  the 
centre  thus  found  to  anyone  of  the  vertices  of  the  tetrahedron. 

PROB.  85.  Take  in  H  one  face  which  we  will  call  the  base. 
Conceive  three  planes  to  be  passed  bisecting  the  basal  dihedral 
angles  of  the  tetrahedron.  A  new  tetrahedron  is  thus  formed 
whose  vertex  is  the  rq.  centre.  To  find  the  auxiliary  solid, 
intersect  both  tetrahedrons  by  a  plane  II  to  H.  By  means  of 
planes  J_  to  the  basal  edges  construct  plane  2£s  measuring 
the  basal  dihedrals  of  the  gn.  tetrahedron  and  revolve  these 
into  H  together  with  the  lines  cut  out  by  the  plane  II  to  H. 
If  through  the  pts.  where  the  bisectors  of  these  £s  pierce  the 
II  plane  lines  be  drawn  II  respectively  to  the  basal  edges  of  the 
tetrahedron,  two  pts.  will  be  known  in  each  of  the  three  lateral 
edges  of  the  auxiliary  tetrahedron,  therefore  its  vertex  (the 
rq.  centre)  will  be  known  and  the  rq.  radius. 

PROB.  86.  In  each  construction  assume  a  face  in  H,  one  of 
whose  sides  is  the  gn.  edge,  o&,  while  no  side  of  the  base  is 
II  or  _L  to  G. 

a)  The  construction  of  the  regular  tetrahedron  presents  no 
difficulty.  In  this  and  succeeding  solids  consider  each  pr.  by 


48  DESCBIPTIVE   GEOMETRY. 

itself  and  use  short  dashes  for  those  prs.  where  the  lines  in 
space  are  hidden  or  covered  by  the  solid. 

b)  It  will  readily  be  seen  that  the  boundary  of  the  H-pr.  of 
the  regular  octahedron  is  a  regular  hexagon,  the  F-pr.  a  paral- 
lelogram. 

c)  Note  first  that  there  are  four  sets  of  vertices,    three  in 
each  set,  at  four  different  altitudes,  found  therefore  in   V  in 
four  different  rows.     Then  note  that  each  set  are  the  vertices 
of  a  regular  A,  that  a  _L  erected  to  the  base  in  H  at  its  middle 
pt.  passes  through  the  centre  of  all  the  A  named  and,  there- 
fore, that  all  the  vertices  fall  in  H  in  two  concentric  circles, 
forming  therein  the  vertices  of  two  similarly  situated  regular 
hexagons.     To  find  the  radius  of  the  larger  circle  note  that  the 
three  vertices  nearest  the  base  are  the  vertices  of  three  regular 
pentagonal  pyramids,  the  bases  of  which  may  be  represented 
in  JET,  each  adjacent  to  a  different  side  of  the  base  of  the  solid. 
From  this  position  revolve  the  pentagons  upon   the  adjacent 
sides  of  the  base  of  the  solid  as  axes  until  the  projected  ver- 
tices of  the  pentagon  nearest  the  triangular  base  meet  in  pairs 
upon  the  produced  medial  lines  of  the  latter.     Any  one  of  the 
pts.  thus  described  determines  the  radius  sought.     The  lowest 
three  vertices  lie  in  H ;  therefore  their  F-prs.  are  in  G.     The 
next  set  are  the  vertices  of  the  pentagonal  pyramids  mentioned 
above.     If  the  altitude  of  the  fundamental  A  be  taken  as  an 
hypotenuse  and  the  altitude  of  the  H-pr.  of  a  A  adjacent  to 
the  base  be  taken  as  the  base  of  a  rt.  A,  the  J_  of  the  latter 
will  be  the  distance  of  the  second  set   of  vertices   above  G. 
The  second  vertical  distance  is  found  in  a  similar  way,  while 
the  fourth  set  of  vertices  are  as  high  above  the  second  as  the 
third  are  above  the  first.     Fig.  19  gives  a  convenient  method 
for  describing  a  regular  pentagon  when  the  length  of  one  side 
AB  is  given.     Give  the  algebraic  proof  that  it  is  correct.     The 
construction  for  the  icosahedron  is  given  in  Fig.  20. 

d}  The  method  of  procedure  with  the  dodecahedron  is  very 
similar  to  that  of  the  icosahedron,  except  the  work  is  somewhat 
simpler.  The  chief  difference  is  found  in  the  fact  that  the 


SUGGESTIONS,   ANALYSES,   AND   DEMONSTRATIONS.     49 

If-pr.  of  the  vertices  of  the  former  are  the  vertices  of  two 
regular  decagons. 

PROB.  87.  The  direction  of  ac  is  found  by  Prob.  20.  The 
direction  of  the  adjacent  edge,  ab  \\  to  H,  can  be  found  by 
Prob.  45,  the  gn.  £  being  an  ^  of  a  regular  pentagon.  These 
two  lines  determine  the  position  of  a  plane  upon  which  an  icos- 
ahedron  satisfying  the  conditions  may  be  placed.  Develop  the 
plane  in  H.  Construct  a  regular  icosahedron  with  upper  face 
in  H,  one  side  of  this  face  being  assumed  II  to  the  H  trace  of 
the  plane.  The  counter-revolution  solves  the  problem.  This 
operation  is  readily  effected  by  utilizing  the  fact  that  the  ver- 
tices lie  in  four  planes  whose  distances  apart  are  known. 

PROB.  89.  Find  the  pts.  where  the  edges  of  the  pyramid 
pierce  the  plane  and  join  these  in  their  proper  order  being 
careful  to  observe  the  rule  with  reference  to  covered  lines,  or 
find  the  lines  in  which  the  planes  of  the  faces  intersect  t'Tt". 
Portions  of  the  lines  form  the  sides  of  mno. 

PROB.  92.  Join  the  vertex  S  with  any  pt.  of  de.  These 
intersecting  lines  determine  an  auxiliary  plane.  If  the  line 
pierces  the  pyramid  and  the  base  of  the  latter  is  in  H,  then  the 
H  trace  of  the  auxiliary  plane  cuts  the  base,  a&c,  in  two  points 
which  are  to  be  joined  to  S.  The  lines  thus  determined  are  the 
section  of  the  auxiliary  plane  and  pyramid  and  cut  the  gn.  line 
in  the  rq.  pts.,  m,  n. 

PROB.  93.  Proceed  in  this  problem  as  in  92,  except  that 
the  auxiliary  plane  is  determined  by  passing  through  any  pt. 
of  de  a  line  II  to  the  lateral  edges  of  the  prism. 

PROB.  94.  The  auxiliary  planes  of  simplest  section  must 
pass  through  both  vertices.  Therefore  determine  the  pt.  in 
which  a  line  joining  the  vertices  S  and  T  pierces  H.  The  H 
traces  of  auxiliary  planes  are  then  passed  through  this  pt.  and 
the  foot  pts.  of  the  edges  of  the  intersecting  surfaces.  If  any 
one  of  these  planes  cuts  both  solids  there  will  lie  in  it  right 
lines  passing  through  S  and  T,  cut  from  both  surfaces,  and 
their  mutual  intersections  will  be  the  salient  pts.  in  the  rq. 
figure,  mno  etc.  Only  those  portions  of  the  intersection  are 


50  DESCRIPTIVE   GEOMETRY. 

to  be  represented  as  visible  in  either  pr.  which  are  visible  in 
each  solid  taken  independently. 

PROB.  95.  Through  any  pt.  in  space  pass  two  lines  II  respec- 
tively to  the  edges  of  abcd-a^c^  and  xyz-x^j^.  The  V 
traces  of  auxiliary  planes  to  be  used  as  in  94  will  be  II  to  the 
V  trace  of  the  plane  of  these  two  lines. 

PROB.  96.  A  construction  similar  to  the  one  required  is 
given  in  Fig.  21.  A  line  is  passed  through  the  vertex  S 
parallel  to  the  lateral  edges  of  the  prism.  The  £Tand  V  traces 
of  the  rq.  auxiliary  planes  pass  through  li  and  v,  respectively,  of 
this  line. 

PROB.  97.  Put  the  plane  in  position  and  construct  unit 
rectangular  axes  ||  to  G^  G3,  G.2,  respectively.  From  the  ex- 
tremities of  these  axes  let  fall  Js  upon  the  plane  and  find  the 
pts.  of  piercing.  Develop  the  plane  together  with  these  pts. 
into  H. 

PROB.  98.  Every  face  of  a  regular  dodecahedron  is  pierced 
at  its  vertices  by  five  edges  of  the  solid,  one  at  each  vertex. 
If  these  edges  be  produced,  they  meet  in  vertices  of  regular 
pentagonal  pyramids  over  the  centres  of  the  several  faces. 
The  entire  figure  thus  derived  is  the  star-dodecahedron  and 
the  new  vertices  are  the  vertices  of  a  regular  icosahedron. 
In  a  similar  way  the  regular  dodecahedron  may  be  formed 
about  the  icosahedron.  These  two  solids  are  therefore  called 
reciprocal.  To  project  the  star-dodecahedron  as  rq.,  the  di- 
rections of  the  axes  are  first  found  by  assuming  oz  in  any 
convenient  position.  Lay  off,  then,  from  o  two  axes  whose 
directions  make  with  oz  ^s  =  to  90°  —  arc  tan  (y^)  for  ox 
and  90°  —  arc  tan  (\)  for  oy.  Construct  the  dodecahedron 
orthographically  with  one  edge  ||  to  G.  The  vertices  of  the 
rq.  figure  are  found  by  taking  the  z  distances  without  change  ; 
while  for  the  new  x  and  y  distances  T\  and  |,  respectively,  of 
the  old  distances  are  taken. 

Fig.  22  shows  the  axonometrical  pr.  of  a  cube  in  the  system 
given  above.  The  meaning  of  this  pr.  is  this  :  a  plane  is 
inclined  to  the  regular  planes  of  pr.,  as  in  84,  and  is  so  taken 


SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS.      51 

that  the  orthographic  pr.  upon  it  of  any  figure  in  its  simple 
descriptive  position  closely  resembles  central  perspective  and 
for  mechanical  purposes  can  be  used  in  place  of  the  latter. 
The  most  common  form  is  the  so-called  isometric  pr.  In  this 
the  unit  rectangular  axes  remain  equal  lines  and  make  angles 
with  each  other  of  120°.  Figs.  1,  3,  5  are  examples  of  its  use. 

The  six  exercises  which  follow  are  closely  connected  with  the 
branch  of  Descriptive  Geometr}T  here  referred  to  and  may  be 
employed  in  determining  the  positions  and  relative  projected 
lengths  of  unit  rectangular  axes  under  given  conditions. 

PROB.  99.  On  any  side  of  the  A  abc,  in  Fig.  23,  as  be,  lay 
off  the  A  a'bc  similar  to  the  A  mno  and  draw  the  circle  K 
passing  through  aa'  and  having  its  centre  in  be.  K  cuts  be  in 
two  pts.,  d',  e',  which  form  with  a  and  a'  two  rt.  A,  dae 
and  da'e,  whose  acute  angles  are  of  different  magnitudes.  Let 
the  2£  aed  be  greater  than  the  ^  a'ed'.  We  take  ae  as  the  axis 
of  affinity  E,  assume  the  projecting  rays  _L  to  E,  lay  off  the 

Y  aed'  =  a  in  a'ed  and  take  the  ratio  of  foreshortening  = 

ad 

In  the  pr.  system  so  determined  the  A  ab'c',  corresponding  to 
the  A  abc,  is  the  A  rq.  For  the  rt.  A  a'ed  and  aed'  are 
similar ;  therefore  the  ^  a'de  =  "$.  ad'e  and  we  have  the  con- 
tinued proportion  :  a'd  :  de  :  dc  :  db  =  ad' :  d'e  :  d'c' :  d'b'.  There- 
fore the  A  a'de  and  ad'e  are  similar ;  hence  a' be  and  ab'c'  are 
similar. 

PROB.  100.  The  solution  is  shown  in  Fig.  24.  Let  a'b'c'  be 
the  given  H-pr.  of  the  A  abc.  Upon  its  base  b'c'  construct  db'c' 
similar  to  mno.  With  centre  in  b'c'  construct  the  circle  K 
passing  through  a'  and  d  and  intersecting  b'c'  in  e  and/.  Com- 
plete the  right  A  a'fe  and  dfe.  Conceive/  to  move  in  a  line_L 
to  H,  a'e  being  taken  as  the  axis  of  affinity,  E  ;  b'  and  c',  as  ver- 
tices of  the  A  a'b'c',  will  also  move  in  a  line  _L  to  H,  while  the 
same  pts.,  as  vertices  of  the  A  db'c',  will  remain  fixed.  When 
a'fe  is  similar  to  dfe,  then  will  abc  in  space  be  similar  to  db'c'. 
Therefore  construct  the  A  a^/  upon  a^  =  a'e  and  similar  to 
def;  then  will  aj\  be  the  hypotenuse  and  aj'^  =  a'f  the  base 


52  DESCRIPTIVE  GEOMETRY. 

of  a  rt.  A  whose  perpendicular  /n/m  will  be  the  altitude 
above  H  of  the  vertex  /of  the  A  afc  when  a  remains  in  H. 
Thence  can  be  easily  deduced  the  corresponding  altitudes  of 
b  and  c. 

PROB.  101.  The  proof  for  the  following  method  is  by 
quaternions  and  ma}'  be  found  in  the  seventeenth  volume  of 
"  Zeitschrift  fur  mathematischen  und  naturwissenschaftlichen 
Unterricht,"  page  481.  In  Fig.  25,  let  0"A",  0"B",  be  the 
V-pr.  of  two  concurrent  edges  of  a  cube.  Make  0"A"M  simi- 
lar to  0"A"B"  and  complete  the  O  0"MNB".  It  will  be 
noticed  that  0"A"  =  ^/0"B"  x  0"M.  Lay  off  0"P=  to  0"N, 
but  in  the  opposite  direction.  Bisect  PO"B"  and  lay  off  0"C" 
=  VO"P  X  0"B".  0"C"  will  be  the  quantity  sought.  It  will 
be  observed  that  these  axes,  when  taken  positively  and  nega- 
tively through  0",  form  the  concurrent  edges  of  eight  equal 
cubes. 

PROBS.  102,  103  and  104.  These  are  instructive  varia- 
tions of  101. 

SECTION  II. 
Lines  and  Surfaces  of  an  Order  higher  than  the  First. 

PROB.  171.  a)  For  the  ellipse  take  in  Fig.  26  the  medial 
lines  AA',  BB',  of  the  O  as  axes.  Draw  lines  CN  II  to  the 
diagonal  DO.  Connect  pts.  C  with  A,  pts.  N  with  A.  The  in- 
tersections P  of  corresponding  lines  are  pts.  upon  the  rq.  in- 
scribed ellipse  and  BB',  AA,  are  conjugate  diameters.  For, 
draw  PM,  PF,  II  respectively  to  BB',  AA. 

Let       OA  =  a,  OB  =  b,  OM=  x,  OF  =  y ; 
ON      CD       ,.ON=qb,CD  =  qa. 


qa      a  —  x 


SUGGESTIONS,  ANALYSES,  AND   DEMONSTRATIONS.      53 
Multiplying  the  first  result  by  the  second,  member  for  member, 


a2      a2  —  or 
the  well-known  equation  of  the  ellipse. 

&)  For  the  hyperbola  take  axes  in  Fig.  27  as  in  the  ellipse, 
draw  lines  NN*  II  to  the  diagonal  OE  ;  through  the  pts.  N1  draw 
lines  N'C  \\  to  the  other  diagonal  OD.  The  intersections  P  of 
lines  A'N  with,  their  corresponding  lines  AC  are  pts.  of  an  hy- 
perbola ;  for,  let  OA  =  a,  OB=  6,  OM=x,  OF=y, 


uu 

A.M 

qa 

x  —  a 

DA 
A'O 

MP 
AM 

b 
a 

y 

ON 

MP 

qb 

y 

Multiplying  these  two  equations  together  member  for  member, 

a2      x2  -  a2  . 
we  have   -=_-^_, 

whence  a2y2  —  tfa?  =  —  a262,  the  equation  of  the  x  hyperbola. 
c)  For  the  parabola  take  0  in  Fig.  28  for  the  origin  and  OX 
and  OY  for  axes.  Draw  AM  and  ||  to  AM,  right  lines  NE. 
Through  the  pts.  N  draw  lines  ||  to  OX;  join  pts.  E  with  O. 
The  intersections  P  are  pts.  of  the  parabola  passing  through 
O,  C,  D,  and  having  OX  for  a  diameter  and  0  F  f  or  a  tangent 
at  the  vertex  0. 

Let  0^1  =  ft,   0^tf=a,  FP=  qb,  AE  =  qa  ;  —  =  -5^. 


=y  and  OF=x;  then^  =  —  ,  also,  y  =  ob  : 

x      qa 

b2 
by  multiplication  2/2  =  —  #,  the  equation  of  the  parabola. 

(X 

PROB.  172.  a)  Let  Fig.  29  represent  a  circle  and  circum- 
scribed square  :  we  wish  to  find  pairs  of  pts.  in  the  square  which 
determine  rectilinear  tangents  to  the  circle.  Let  a  tangent  with 
the  pt.  of  tangency  Jc  cut  ab  and  ad  in  the  pts.  p  and  q. 


54  DESCRIPTIVE  GEOMETRY. 

Let       up  =  #,     qa  =  y. 

In  the  rt.  A  apq  we  have 

pa  =  r  —  x  (r  =  radius)  ,  pq  =  pk  +  kq  =  x  +  r  —  y, 
p(f—pa  -f  qa    or  =  (x  +  r  —  y)2  =  (r  —  x)2  +  y2. 


r  +  x 

Draw  the  chord  su  ;  through  p  draw  the  parallel  to  uv,  cutting 
su  in  r  ;  then  draw  br  which  cuts  ad  in  some  pt.  q'  whose  dis- 
tance from  a  we  will  designate  by  y'.  .  .  .  Then  because 

bp  :  pr:  :  ab  :  aq' 

or  r  +  x  :   x  :  2r  :  :  y\ 

we  have  y1  —  —    —•     .-.  q  falls  upon  q'. 
r  -f-  x 

For  the  pt.  of  tangency,  &,  pk  =  pu,  also  mp  and  vk  are  _L  to 
wfc  and  .•.  II,  .-.  the  intersection,  ^,  of  vk  and  ms  lies  upon  pr. 
The  parallel  projection  of  a  circle  is  an  ellipse,  while  parallels 
will  remain  parallel,  tangents  will  remain  tangents,  etc.  There- 
fore the  parallel  projection  of  Fig.  29  will  give  the  construc- 
tion in  the  upper  left-hand  quadrant  of  Fig.  26  for  an  ellipse 
determined  as  the  envelope  of  a  system  of  tangents. 

b)  If  a  A  of  constant  area  be  cut  off  by  a  moving  line  from 
two  fixed  intersecting  lines,  we  know  from  Analytic  Geometry 
that  the  envelope  of  the  moving  line  is  an  hyperbola  and  that  the 
middle  pt.  of  the  line  is  in  every  position  the  pt.  of  tangency. 
Taking  the  intersecting  lines  as  axes,  the  included  angle  as  <£, 
K  as  the  constant  area  of  the  A  and  x  and  y  as  the  intercepts 
upon  the  axes,  we  have  for  the  middle  pt.  of  the  moving  line 
\x-\y-  sin  <£  =  2k.       .-.    xy  =  constant,  the    equation   of    an 
hyperbola  referred  to  its  asymptotes. 

c)  If  two  intersecting  lines  are  divided  into  equal  parts  the 
pts.   on  one   line  beginning  at  the  pt.  of   intersection   being 
numbered  from  1  to  n,  on  the  other  from  n  to  1,  and  like  num- 
bers are  joined  by  right  lines,  there  will  be  formed  the  envelop- 
ing tangents  of   a  parabola.      The  proof   is  from  Projective 
Geometry. 


SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS.      55 

PROB.  173.  When  the  principal  axes,  AA\  BB',  are  given, 
and  circles  are  described  about  them  as  diameters,  a  pt.  upon 
the  ellipse  may  be  found  as  follows  :  Draw  any  line  from  the 
common  centre  intersecting  both  circles.  From  the  outer  pt.  of 
intersection  let  fall  a  J_  to  the  major  axis,  from  the  inner  a  _L 
to  the  minor  axis.  The  pt.  of  intersection  of  the  J§  is  the  rq. 
pt.  For  construction,  see  Fig.  30.  When  conjugate  diameters 
are  given,  the  method  by  circles  is  shown  in  Fig.  31. 

PROB.  174.  The  construction  of  this  problem  is  shown  in 
Fig.  32.  It  is  from  Parallel  Projection,  and  is  known  as  the 
Shadow  Method.  Let  aa'  and  /?/3'  be  the  gn.  conjugate  axes. 
Through  ft  draw  E  II  to  aa'.  E  must  be  tangent  to  the  ellipse 
whose  axes  are  required  and  is  taken  as  an  axis  of  affinit}-. 
From  /?  erect  pc  J_  to  E  and  =  4-  aa'  =  ao.  With  c  as  centre 
and  c/3  as  radius,  describe  the  circle  K.  The  ellipse  K±  is  the 
II  pr.  of  K.  All  pairs  of  diameters  _L  to  each  other  in  K  will 
be  projected  as  conjugate  diameters  in  K^  The  pair  of  _L 
diameters  of  K  which  remain  _L  in  ^i,  are  the  principal  diam- 
eters of  the  ellipse.  Pass  through  o  and  c  a  circle,  K',  with 
centre  in  E.  The  J_  diameters  aa',  66',  of  K  remain  _L  in  K^ 
and  are  in  projection  the  principal  diameters  AA  ',  BB',  sought. 

PROB.  175.  It  is  a  fundamental  proposition  in  transversals 
that  if  a  right  line  intersect  the  three  sides  AB,  BC,  CA,  of  a 
A  ABC,  or  those  sides  produced  in  pts.  JF\,  P2,  P3,  respec- 
tively, then  the  product  of  three  non-adjacent  segments  equals 
the  product  of  the  other  three  ;  or 


and  conversely,  when  the  above  relation  holds,  Px,  P2,  P3,  are 
in  one  right  line.  Starting  from  this  proposition,  we  shall  prove 
Pascal's  proposition  applying  to  his  so-called  Hexagrammatica 
Mystica.  The  proposition  is  this  :  if  six  pts.  lie  on  the  cir- 
cumference of  a  circle  and  an  inscribed  convex  or  re-entrant 
hexagon  be  formed  with  these  pts.  as  vertices,  then  the  opposite 
sides,  or  sides  produced,  meet  in  three  points  lying  upon  a  right 


56  DESCRIPTIVE   GEOMETRY. 

line  which  we  shall  call  the  axis.     In  the  A  LMN  of  Fig,  J3, 
let  AP,  FQ,  CR,  be  taken  successively  as  transversals. 
We  have 


LA-MB-NP=l      LF>  MQ-NE=l       LR-MC-ND 
AM'BN-LP  MF-NQ.LE  MR.NC>LD~ 

Also  since  LD,  LA,  etc.  are  secants, 

LE-LD  =  1      MA-MF==1      NC-NB 
LF-LA  MB-MC  ND-NE~ 

Multiplying  these  six  equations  together  we  have 


LP-NQ.MR 

By  the  converse  of  the  proposition  regarding  transversals, 
P,  Q,  R,  must  be  in  the  same  right  line.  Since  the  ellipse, 
hyperbola  and  parabola  are  formed  by  the  central  projection  of 
the  circle  and  since  in  such  pr.  rt.  lines,  tangents,  pts.  of 
intersection  and  pts.  of  tangency  in  the  projected  figure  re- 
main the  same  in  the  projection,  it  follows  that  the  demonstra- 
tion gn.  applies  to  the  curves  of  the  second  degree. 

If  five  pts.,  A,  .B,  (7,  D,  E,  are  gn.  in  one  plane,  any  num- 
ber of  lines  AF  may  be  drawn,  and  the  corresponding  pts. 
F  may  be  found  by  using  in  each  case  the  corresponding  axis. 
Therefore,  when  five  pts.  are  gn.  in  a  plane,  one  conic  section 
can  always  be  found  containing  them  and  only  one.  An 
inscribed  pentagon  may  be  regarded  as  an  inscribed  hexagon 
in  which  one  pair  of  adjacent  vertices  are  consecutive  ;  an 
inscribed  quadrilateral,  one  in  which  two  pairs  are  so  related  ; 
an  inscribed  triangle,  one  in  which  three  pairs  coincide.  In  the 
last  three  cases  a  pair  of  consecutive  vertices  are  not  fully  gn. 
unless  the  direction  of  the  line  joining  them  is  gn.  This  line  is 
tangent  to  the  conic  section  in  which  the  consecutive  vertices 
lie.  If  the  five  pts.  assumed  form  a  convex  figure,  the  conic 
section  containing  them  will  be  an  ellipse,  as  in  a)  of  this 
Prob.  If  the  five  pts.  assumed  form  a  re-entrant  figure,  the 
section  will  be  an  hyperbola,  as  in  a)  of  176.  If  three  pts.  be 


SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS.      57 

assumed  and  a  direction  leading  at  infinity  to  two  other  pts., 
and  if  the  latter  are  consecutive  with  the  line  joining  them, 
the  infinite  right  line  of  the  plane,  then  the  section  will  be  a 
parabola,  as  in  177. 

The  proposition  of  Brianchon  is  derived  from  that  of  Pascal 
by  the  application  of  the  principle  of  pole  and  polar.  It  is  as 
follows  :  If  a  hexagon  be  circumscribed  to  any  one  of  the  conic 
sections,  the  three  diagonals  joining  opposite  vertices  will  pass 
through  the  same  pt.  In  Pascal's  hexagram  it  is  convenient  to 
arrange  the  symbols  for  the  lines  in  a  row  and  connect  them  as 
rq.  If  A,  B,  C,  D,  E,  F,  be  the  vertices,  then  AB,  BC,  CD, 
DE,  EF,  FA,  will  be  the  sides,  and  AB  -  DE,  BC  -  EF,  CD  .  FA, 
will  be  the  three  collinear  intersections. 

In  Brianchon's  hexagon  a,  6,  c,  d,  e,  /,  will  be  the  vertices 
and  ad,  be,  c/,  the  lines  which  intersect  in  a  pt.  It  is  evident 
that  in  the  former,  when  five  vertices  are  gn.  any  number  of  sixth 
vertices  belonging  to  the  same  conic  section  can  be  found.  In 
the  latter,  five  tangents  being  gn.  any  sixth  tangent  can  be  found. 

Two  pts.  each  of  a)  and  b)  are  constructed  in  Figs.  34  and 
35.  In  Fig.  34,  the  axis  always  passes  through  P.  In  finding 
/  it  also  passes  through  the  intersection  of  be  and  the  assumed 
direction  ef.  Similarly  for  ef.  In  Fig.  35,  A,  B,  etc.,  denote 
the  sides  of  the  gn.  pentagon  ;  the  intersection  of  A  and  B 
is  denoted  by  A  •  B,  of  B  and  C,  by  B  -  C,  etc.  Then  the 
order  for  finding  any  number  of  tangents,  F,  is 

A -B,  B^C,  C'-D,  D-E,E',F,F\  A. 

Two  tangents,  F,  are  determined  in  Fig.  35.  Determine  the 
principal  axis  by  174. 

PROB.  177.  Two  pts.  of  the  rq.  parabola  are  constructed  in 
Fig.  36.  ab  is  produced  to  intersect  at  oo  with  the  infinite 
right  line  de  ;  the  latter  is  the  direction  to  ab.  The  line  f'e 
is  assumed.  Its  intersection  with  be  determines  one  pt.  of  E< 
E  being  drawn  to  the  intersection  of  ab  with  de  must  be  II  to  ab, 

PROB.  179.  Use  as  an  auxiliary  line  the  element  upon  whicl^ 
a  is  found. 


58  DESCRIPTIVE    GEOMETRY. 

PROB.  181.  Use  the  plane  _L  to  ool  at  o  as  an  auxiliary  sur- 
face and  its  traces  as  axes  of  affinity  between  the  elliptical 
projections  of  the  base  and  its  circular  developments. 

PROB.  183.  The  development  is  theoretically  effected  by 
opening  the  cone  along  some  element  and  rolling  the  surface 
out  into  a  plane.  It  is  practically  done  by  dividing  the  base 
into  any  number  of  equal  parts.  The  true  lengths  of  the  ele- 
ments passing  through  these  pts.  of  division  are  found  as  nearly 
as  practicable  and  the  true  length  of  the  equal  arcs.  These 
constituent  lines  combined  in  the  proper  order  form  triangles, 
which,  placed  adjacent  in  a  plane  in  the  same  relation  in  which 
they  stand  in  the  surface,  are  the  development  of  the  latter. 
A  continuous  curve  passed  through  the  non-concurrent  ends  of 
the  developed  elements  is  the  developed  base.  This  work 
requires  the  rectification  of  the  circle.  Three  and  one-seventh 
times  the  diameter  is  0.0013  of  the  circumference  too  much. 
Three  times  the  diameter  increased  by  one-fifth  the  chord  of  a 
quadrant  is  0.0003  of  the  circumference  too  small.  Both  of 
these  errors  are  smaller  than  those  that  ordinarily  arise  from  the 
use  of  instruments. 

When  the  base  of  the  cone  is  a  curve  not  easily  rectified,  it 
is  usually  accurate  enough  for  descriptive  purposes  to  take 
divisions  of  the  base  so  small  that  the  arcs  shall  not  sensibly 
differ  from  their  chords.  Occasionally  the  development  is 
effected  by  using  the  intersection  of  the  cone  with  a  sphere 
whose  centre  is  at  the  vertex.  The  developed  intersection  is 
then  the  arc  of  a  circle. 

PROB.  185.  By  Analytic  Geometry  it  is  proven  that  the  in- 
tersection of  a  plane  with  a  cone  of  the  second  order  is 

a)  an  ellipse,  when  it  cuts  all  the  elements  ; 

b)  a  parabola,  when  it  is  parallel  to  one  and  only  one 

element ; 

c)  an  hyperbola,  when  it  is  parallel  to  two  elements. 
Take  t'Tt"  so  as  to  give  a),  6),  and  c),  successively. 
PROB.   187.    Similar  to  101. 

PROB.   188.    Similar  to  100, 


SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS.      59 

PROB.  189.  Revolve  the  line  upon  its  ZT-pr.  into  H  together 
with  the  small  circle  cut  out  from  the  sphere  by  its  H  project- 
ing plane. 

PROB.  190.  In  elementary  Geometry  it  is  shown  that  the  line 
cut  out  from  the  plane  of  the  base  of  the  cone  by  the  tangent 
plane — in  other  words  the  V  trace  of  the  latter  as  here  given 
—  must  be  tangent  to  the  base  of  the  cone  at  the  foot  of  the 
element  containing  p.  Tt"  and  the  pt.  where  a  line  from  JS  or 
any  pt.  of  the  element  of  tangency  and  II  to  Tt"  pierces  H, 
will  determine  t'T. 

PROB.  192.  Draw  an  auxiliary  line  through  S  and  p  and  find 
its  h.  t'Tmust  pass  through  li  and  be  tangent  to  the  base  of 
the  cone.  In  general,  there  are  two  solutions  for  cones  of  the 
second  order. 

PROB.  193.  The  auxiliary  line  is  drawn  through  p  II  to  the 
elements  of  the  cylinder.  Otherwise  as  in  192. 

PROB.  196.  Pass  an  auxiliary  line  through  S  II  to  the  direc- 
tion mn,  which  is  a  gn.  line. 

PROB.  198.  Construct  an  auxiliary  cone  with  vertex  at  any 
pt.,  p,  and  with  elements  making  with  H  the  £  e.  Through  p 
pass  a  line  II  to  the  elements  of  the  cylinder.  The  tangent 
plane  upon  the  auxiliary  cone  determined  by  this  line  will  be  II 
to  the  plane  rq. 

PROB.  199.  Construct  an  auxiliary  cone  of  revolution  with 
its  vertex  at  the  vertex  of  the  cone  and  its  elements  making 
the  ^  c  with  H.  The  common  tangent  plane  will  be  the  plane 
rq. 

PROB.  200.  The  helix  is  denned  in  Part  III.,  Art.  27,  and 
is  constructed  in  Fig.  37. 

The  developable  or  tangential  surface  is  defined  in  Part  III., 
Art.  46,  and  constructed  upon  the  helix  of  Fig.  37. 

When  the  helix  with  axis  JL  to  H,  forms  the  directrix  as  in 
this  case,  the  base  of  the  surface  in  H  is  the  involute  of  the 
circle  forming  the  //-pr.  of  the  helix.  If  a  rt.  A  be  cut 
from  paper  and  wound  about  a  cylinder  with  one  of  its  legs  II 
to  the  axis  of  the  latter,  the  hypotenuse  will  be  very  nearly  a 


60  DESCRIPTIVE   GEOMETRY. 

helix.  Unwind  the  paper,  keeping  the  unwound  portion  plane. 
The  hypotenuse  will  approximately  describe  a  developable  sur- 
face and  the  vertex  at  the  lower  acute  angle  the  involute  base. 
The  tangent  plane  is  drawn  as  in  the  cone. 

PROB.  201.  Take  the  axis  of  the  helical  directrix  J_  to  H. 
With  any  pt.  of  ab  as  a  vertex,  describe  an  auxiliary  cone  of 
revolution  whose  elements  make  with  H  the  same  ^  as  the  ele- 
ments of  the  gn.  surface.  The  plane  containing  ab  and  tan- 
gent to  the  auxiliary  cone  will  be  II  to  t'Tl". 

PROB.  202.    Take  auxiliary  planes  as  in  103. 

PROB.  203.  The  problem  is  similar  to  96.  The  auxiliary 
planes  will  be  II  to  a  plane  determined  by  two  intersecting  lines  ; 
one  II  to  the  elements  of  the  cylinder,  the  other  to  the  edges  of 
the  prism. 

PROB.  205.  Auxiliary  planes  cutting  rectilinear  elements 
from  both  surfaces  must  contain  the  line  passed  through  S 
II  to  the  elements  of  K.  In  other  words,  they  must  pass 
through  the  v  and  h  of  this  line.  Especial  attention  should  be 
given  to  the  auxiliary  planes  whose  traces  are  tangent  to  the 
bases  of  one  or  both  of  the  surfaces,  for  such  planes  contain 
elements  which  are  tangent  to  the  line  of  intersection.  The 
solution  is  similar  to  Fig.  21. 

PROB.  208.    t'Tt"  will  be  X  to  the  radius  of  tangency  at  p. 

PROB.  209.  Construct  auxiliary  cones  of  revolution  tangent 
to  the  sphere  and  with  vertices  in  v  and  h  of  ab.  The  planes 
of  the  bases  of  these  cones  will  be  the  corresponding  project- 
ing planes  of  the  chord  of  the  sphere,  connecting  the  two  pos- 
sible pts.  of  tangency.  By  revolving  either  projecting  plane 
into  its  corresponding  plane  of  pr.  and  with  it  the  small  circle 
cut  by  it  from  the  sphere,  the  pts.  of  tangency  are  found. 

PROB.  210.  Pass  a  plane  through  c  _L  to  ab  and  determine 
the  pt.  cut  from  ab.  Develop  into  either  H  or  Fthis  pt.  and 
the  great  circle  cut  from  the  sphere.  The  tangent  from  pt.  to 
circle  is  the  developed  position  of  a  tangent  intersecting  ab 
and  with  it  determining  t'Tt".  Two  solutions. 

PROB.  211.   An  hyperboloid  of  revolution  of  one  nappe  or 


SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS.      61 

sheet  is  generated  by  the  revolution  of  an  hyperbola  about  its 
imaginary  axis.  It  is  also  generated  by  the  revolution  of  one 
right  line  about  another  to  which  it  is  windschief  and  with 
respect  to  which  its  relation  is  constant.  To  prove  the  surface 
thus  generated  is  an  hyperboloid  of  one  sheet,  it  is  only  neces- 
sary to  prove  that  its  meridian  section  is  an  hyperbola  whose 
imaginary  axis  is  the  axis  of  revolution. 

In  Fig.  38,  let  v  be  any  pt.  in  the  meridian  line,  t"m"v".  Let 
ab  be  an  element  II  to  V.  Let  o'm'  =  o'd'  =  d"m"  =a  =  the 
radius  of  the  circle  of  the  gorge.  In  the  rt.  A  d"m"c",  let  m"c" 
=  b.  Let  r'v'  =  x  =  r's'  =  r"v",  and  d"r"  =  y. 

In  the  rt.  A  r'd's' 


vQ  r*Y>  n 

but  d's'  =  r"s",  and  LJL  =  CL^_,  Or    d's'  =  %. 

r"d'      m"c"  b 

n 

.  •  .  x2  =  a2  H  --  ?/2,  or  a?y2  —  tfx2  =  —  a262, 
62 

the  equation  of  the  x  hyperbola.     Read  Part  III.,  Arts.  53-62. 

The  revolution  of  the  common  _L  generates  the  smallest  par- 
allel of  the  surface.  This  parallel  is  called  the  circle  of  the 
gorge. 

A  right  line  intersecting  the  generatrix  at  the  foot  of  the 
common  _L  from  generatrix  to  axis  and  making  with  the  latter 
the  same  angle  in  space,  will  evidently  generate  the  same  sur- 
face, since  every  pt.  of  the  new  generatrix  will  be  at  the  same 
distance  from  the  axis  as  its  homologous  pt.  in  the  old  genera- 
trix. Therefore  through  any  pt.  of  the  surface  there  will  pass 
two  rectilinear  elements,  one  from  each  generation,  and  these 
determine  the  tangent  plane  at  that  point.  It  is  also  easily 
seen  that  an  element  of  one  generation  cuts  every  element  of 
the  other. 

Let  the  hyperboloid  be  cut  by  two  planes  _L  to  the  axis  and 
at  equal  distances  above  and  below  the  plane  of  the  gorge.  If 
the  axis  is  J_  to  H,  an  element  joins  pts.  in  these  bases  whose 
.H-prs.  are  separated  by  an  arc  greater  than  zero  and  less  than 


62  DESCRIPTIVE   GEOMETKY. 

180°.  The  tangents  at  these  two  pts.  are  therefore  not  II,  and, 
since  they  do  not  intersect,  they  are  windschief .  An  element 
and  its  successive  element,  therefore,  are  lines  connecting 
different  pts.  of-  two  windschief  lines  and  must  themselves, 
therefore,  be  windschief.  Hence  the  surface  itself  is  warped. 
See  Part  III.,  Art.  44,  3. 

PROB.  212.  The  tangent  plane  is  determined  most  readily  by 
two  tangents  through  p,  one  to  the  meridian  section,  the  other 
to  the  parallel  through  the  gn.  pt. 

PROB.  214.  If  with  the  given  line  as  generatrix  and  the  axis 
of  the  given  solid  as  axis,  an  hyperboloid  of  revolution  of  one 
sheet  be  formed,  the  tangent  plane  will  be  tangent  to  this 
auxiliary  surface  at  some  pt.  of  the  given  line  ;  for  a  plane  con- 
taining an  element  of  a  warped  surface  must  be  tangent  to  that 
surface  at  some  pt.  of  the  element.  See  Part  III.,  Art.  66. 

Also,  the  meridian  plane  passed  through  the  pt.  of  tangency 
of  the  given  solid  must  cut  out  a  common  tangent  from  the 
common  tangent  plane.  If,  then,  we  lay  in  T^the  common  tan- 
gent upon  the  bounding  lines  of  the  two  surfaces  of  revolution, 
and  determine  the  pt.  of  tangency  upon  the  hyperboloid,  that 
pt.  will  give  the  parallel  circle  containing  the  pt.  of  the  given 
line  where  the  latter  is  cut  by  the  common  meridional  tangent. 
This  tangent,  with  the  given  line,  determines  the  rq.  plane. 

PROB.  215.  Pass  a  system  of  planes  through  p  and  the  ellip- 
soid and  to  each  section  determine  the  tangent  line  and  the  pts. 
of  tangency.  The  former  will  be  the  elements  of  the  rq.  cone  ; 
the  latter,  pts.  in  the  rq.  curve  of  contact.  This  curve  is  plane 
and  of  the  2d  order. 

PROB.  216.  Use  the  projecting  planes  of  the  cylindrical 
elements  as  auxiliary  surfaces.  In  general,  they  cut  right  lines 
from  the  cylinder  and  small  circles  from  the  sphere  whose 
mutual  intersection  is  determined  by  revolution. 

PROB.  217.  For  auxiliary  surfaces  take  spheres  with  their 
common  centre  at  the  intersection  of  the  axes. 

PROB.  218.  Assume  an  axis  _L  to  //  whose  JJ-pr.  is  the 
middle  pt.  of  an  ellipse  assumed  as  the  basal  section  and  the 


SUGGESTIONS,  ANALYSES,  AND  DEMONSTRATIONS.      63 

projection  of  the  upper  base.  Lay  off  at  will  in  F,  along  the 
axis  from  6r,  any  distance  as  the  altitude  of  the  bounded  solid. 
The  plane  of  the  ellipse  of  the  gorge  bisects  this  distance.  The 
JET-pr.  of  the  gorge  may  be  assumed,  in  which  case  it  is  concen- 
tric with  the  base  and  similar  to  it.  The  hyperboloid  of  revo- 
lution is  best  represented  by  dividing  the  circular  base  into  any 
number  of  equal  parts  and  drawing  elements  from  the  pts.  of 
division.  In  the  general  hyperboloid  under  discussion,  the  foot 
points  of  elements  corresponding  to  these  may  be  found  by 
describing  a  circle  about  the  major  axis,  dividing  this  circle  into 
any  number  of  equal  parts  and  projecting  the  pts.  of  division 
upon  the  ellipse  by  lines  J_  to  the  major  axis.  This  construc- 
tion is  seen  to  resemble  the  method  of  Fig.  30  and  is  in  reality 
dependent  upon  the  orthographical  projection  of  a  circle  whose 
plane  is  oblique  to  the  plane  of  projection.  A  pt.  in  the  base 
of  the  asymptote  cone  is  found  by  passing  through  the  axis  a 
plane  parallel  to  any  element  and  projecting  its  foot  pt.  upon 
this  plane.  Read  also  Part  III.,  Arts.  63-74. 

PROB.  219.  Without  entering  further  into  the  discussion  of 
this  surface,  it  may  simply  be  stated  that  it  is  one  of  the  limiting 
cases  of  the  hyperboloid  of  one  nappe  and,  like  that  surface, 
has  a  double  system  of  generation. 

To  represent  the  surface  symmetrically,  divide  portions  of 
two  windschief  lines  into  the  same  number  of  equal  parts  and 
join  homologous  pts.  Any  two  elements  of  one  generation 
may  be  taken  as  the  directrices  of  the  second  with  which  the 
operation  given  above  may  be  repeated,  as.  shown  in  Fig.  39. 
Various  beautiful  projections  of  this  interesting  surface  can 
be  easily  contrived. 

The  two  rectilinear  elements  passing  through  any  pt.  of  a  sur- 
face determine  the  tangent  plane  at  that  pt. 

PROB.  220.  Through  p  and  cd  pass  a  cone  intersecting  the 
projecting  cylinder  of  ef  in  a  curve,  xy,  whose  actual  intersec- 
tion with  ef  determines  a  second  pt.,  g,  in  the  rq.  element. 
The  construction  is  given  in  Fig.  40. 

PROB.  221.    For  case  a),  construct  a  series  of  lines  in  the 


64  DESCRIPTIVE   GEOMETKY. 

plane  directer.  Through  p  pass  a  cone  of  rays  II  to  this  series 
and  find  a  second  pt.  in  the  rq.  element,  as  in  220. 

For  case  5),  pass  through  ab  a  cylinder  of  rays,  each  of 
which  is  parallel  to  mn.  Find  a  second  pt.  as  in  230. 

PROB.  222.  Construct  the  helical  directrix  with  axis  _L  to  H. 
Divide  the  circular  base  into  any  number  of  equal  parts.  Draw 
one  of  the  elements  lying  II  to  Fat  the  gn.  %.  in  V  with  a"6", 
and  intersecting  the  latter  in  Sn.  Divide  the  portion  of  the 
axis  in  V  above  and  below  S"  corresponding  to  one  spire  of 
the  helix  into  the  same  number  of  equal  parts  as  the  circular 
base.  Connect  the  successive  pts.  in  the  helix,  above  and 
below  the  element  already  drawn,  with  the  corresponding  pts. 
of  the  axis. 

Assume  a  pt.  in  the  surface  b}*  assuming  one  of  its  prs.  and 
finding  the  other  by  constructing  the  rectilinear  element  upon 
which  it  lies. 

PROB.  223.  To  pass  a  plane  tangent  to  a  helicoid  at  any  pt. 
of  its  surface,  we  construct  the  subordinate  helix  upon  which 
the  pt.  lies  and  the  right  line  tangent  to  this  helix  at  the  gn. 
pt. 

The  constructed  tangent  and  the  rectilinear  element  passing 
through  the  pt.  determine  the  plane  sought. 

PROB.  225.  The  auxiliary  surfaces  most  convenient  for  appli- 
cation are  planes  _L  to  the  axis  of  the  hyperboloid.  The  prin- 
cipal axis  of  the  section  is  the  intersection  of  t'2V  with  that 
meridian  plane  which  is  _L  to  it. 


PART  III. 

SUMMAEY  OF  PKINCIPLES  AND  DEFINITIONS. 

SECTION  I. 
Projections. 

1.  Descriptive  Geometry.      Descriptive    (darstellende,    bes- 
chreibende)   Geometry  is  the  science  and  art  of  the  methods 
by  which  the  form  and  position  of  geometrical  solids  are  rep- 
resented by  drawings  and  by  which   all  constructions   rq.    in 
space  can  be  solved  with  the  help  of  such  elements  as  can  be 
represented  in  a  plane. 

2.  Shading    and    Perspective.      Descriptive    Geometry   in- 
cludes Shading  and  Perspective  and  is,  therefore,  the  foundation 
of  the  art  of  Drawing. 

3.  The  representation  of  a  solid  may  be  effected  in  general  in 
two  ways,  giving  rise  to  Central  Projection  and  Parallel  Projection. 

4.  Central  Projection.     The  Central  Projection  of  a  figure  is 
the  intersection  of  a  plane  and  a  pencil  of  rays. 

The  plane  is  the  Plane  of  Projection,  or  the  Picture  Plane. 
The  pencil  of  rays  is  composed  of  straight  lines,  each  passing 
through  the  fixed  or  central  point  —  called  the  Point  of  Sight — , 
a  point  of  the  figure  and  a  point  of  the  picture  plane. 

5.  Parallel  Projection.     The  Parallel  Projection  of  a  solid  is 
the  intersection  of  a  plane  and  a  system  of  parallel  lines  :  it  is, 
in  fact,  central  projection  with  the  point  of  sight  at  infinity. 

The  plane  is  the  Plane  of  Projection  ;  the  parallel  lines,  Pro- 
jecting Lines.  All  the  projecting  lines  of  a  continuous  figure  in 
space  form  a  Projecting  Cylinder.  For  the  right  line  in  gen- 
eral and  for  special  positions  of  plane  figures,  the  projecting 
cylinder  becomes  a  Projecting  Plane.  Parallel  projection  is 
Oblique  or  Orthogonal. 

6.  Oblique  Parallel  Projection.     In  oblique  parallel  projec- 
tion the  projecting  lines  are  oblique  to  the  plane  of  projection. 


66  DESCRIPTIVE    GEOMETRY. 

7.  Orthogonal  Parallel  Projection.     In  orthogonal  or  ortho- 
graphic parallel  projection  the  projecting  lines  are  perpendicu- 
lar to  the  plane  of  projection. 

8.  In  this  hand-book,  whenever  the  projection  of  a  figure  is 
mentioned,  the  orthographic  is  meant,  unless  otherwise  speci- 
fied, and  Descriptive  Geometry  will  be  taken  to  mean  the  sci- 
ence and  art  of  orthographic  progression. 

9.  Planes  of  Projection.     In  orthographic  projection  usually 
two  principal  planes  of  projection  are  taken  at  right  angles  to 
each  other,  one  the  horizontal,  H\  the  other  the  vertical,    F. 
To  these  may  be  added  a  third,  P,  perpendicular  to  both.     Ex- 
cept in  special  positions  of  the  figure  to  be  projected,  it  is  fully 
determined  without  the  aid  of  P. 

10.  Ground  Line.     The   intersection   of   H  and    V  is   the 
ground  line,  G ;  of  Fand  P,  the  G2 ;  of  P  and  H,  6r3. 

11.  Revolutions.     By  revolution   through   an  angle  of   90° 
about    6r,   H  and    V  are    made    coincident.      To    effect   this 
transformation,  which  is  the  fundamental  one  of  Descriptive 
Geometry,  we  may  suppose  the  part  of  Fabove  G  revolves  back- 
wards, the  part  below  G  forwards  through  90°,  or  that  the  back 
part  of  H  revolves  upwards,  the  fore  part  downwards,  through 
the  same  angle.     For  the  sake  of  uniformity  the  latter  change 
will  be  always  understood,  unless  otherwise  specified.     When 
H  and  V  have  been  made  coincident,  the  part  of  the  composite 
plane  above  G  represents  upper  Fand  back  H;  the  part  below 
G,  front  H  and  lower  V.     Similarly  P  is  revolved  through  90° 
upon  6r2  and  it  will  be  uniformly  assumed  that  the  fore  part  of 
P  is  superposed  upon  the  right-hand  part  of  F. 

SECTION  II. 
Point,  Line  and  Plane. 

12.  Point.     A  point  is  completely  determined  by  its  two 
projections.     The  projections  of  a  point  lie  in  the  same  line  _L 
to  G. 


DEFINITIONS.  67 

13.  Right  Line.     The  projections  of   a  right  line  are  also 
right  lines.     In  general,  a  line  is  completely  determined  by  its 
first  two  projections,  except  when  it  lies  in  a  plane  _1_  to  G. 

14.  Traces  of  a  Line.    The  points  in  which  a  line  pierces  H, 
Fand  P  are  its  traces,  A,  v,  p. 

NOTE.  Statements  will  now  be  confined  to  the  first  two  projections. 
The  student  will  readily  make  and  prove  similar  statements  for  the  third 
projection. 

15.  The  h  and  v  of  a  line  are  in  its  jff-pr.  and  F-pr.,  respec- 
tively, and  are  their  own  projections  of  the  same  name.     Also 
the  F-pr.  of  h  and  the  H-pr.  of  v  are  in  G.      If,  therefore, 
from  the  point  in  which  the  F-pr.  of  the  line  cuts  G  a  _l_  to  G 
be  drawn  in  H,  its  intersection  with  the  H-pr.  of  the  line  is  h : 
similarly  for  v. 

16.  Angle   of  Inclination.     The  angle  of   inclination  of   a 
line  or  plane  is  the  angle  it  makes  with  a  plane  of  projec- 
tion. 

17.  Parallel  Right  Lines.     If   two  right  lines  are  II,  their 
like-named  projections  are  II. 

18.  Plane,  how  represented.     The  position  of  a  plane  is  in 
general  represented  by  its  intersections  with  H  and  F. 

These  intersections  are  called  the  traces  of  the  plane  and 
must  always  meet  G  in  the  same  point. 

19.  A  plane  is  determined  by  three  points  not  in  the  same 
right  line,  by  a  line  and  point  without  it,  by  two  intersecting 
lines  and  by  two  parallel  lines. 

20.  Point  in  Plane.     If  a  point  lies  in  a  plane,  it  lies  in  any 
line  of  that  plane  whose  projections  pass  through  the  projections 
of  the  point. 

21.  Line  in  Plane.     If  a  line  lies  in  a  plane,  its  h  and  v  lie 
respectively  in  the  .ffand  F  traces  of  the  plane. 

22.  Plane  perpendicular  to  H  or  V.     If  a  plane  is  perpen- 
dicular to  Fits  H  trace  is  perpendicular  to  G  and  vice  versa. 

23.  Line  perpendicular  to  Plane.     If  a  line  is  _L  to  a  plane 


68  DESCRIPTIVE   GEOMETRY. 

its  projections  are  _L  to  the  like-named  traces  of  the  plane  and 
conversely. 

24.  Two  Parallel  Planes.     If  two  planes  are  II,  their  homol- 
ogous  traces  are   II.     The  converse  is  true,  except  when  the 
planes  are  II  to  6r,  then  their  third  traces  must  be  II  to  prove  the 
planes  II. 

SECTION  III. 
Line  in  General. 

25.  Line,  how  Generated.      A   line  is  generated,  a)  by  a 
moving  point,  6)  as  the  envelope  of  a  moving  right  line. 

26.  Plane  Curve.     A  line  is  plane  when  four  consecutive 
points  always  remain  in  the  same  plane. 

27.  Space  Curve.     A  line  is  a  space  curve  when  four  consec- 
utive points  do  not  generally  remain  in  the  same  plane. 

28.  Plane  Curves. 

1.  Algebraical.    The  locus  of  a  rational  algebraical  equation, 

in  general  of  two  unknown  quantities. 

a)  Order.    Number  of  times  intersected  by  a  right  line. 

First  Order.    Right   line  (a  circle  of  infinite  ra- 
dius and  thus  a  curve). 
Second  Order.    Ellipse,  parabola,  hyperbola. 

b)  Class.    Number  of  tangents  that  can  be  drawn  to  it 

from  any  point  in  its  plane. 
First  Class.    Point. 
Second  Class.    Ellipse,  parabola,  hyperbola. 

2.  Transcendental.    The  locus  of  a  transcendental  equation ; 

either    trigonometrical,    circular,    logarithmic,    or    ex- 
ponential. 

29.  Higher  Plane  Curves.     All  plane  transcendental  curves 
and  plane  curves  of  a  higher  order  or  class  than  the  second,  are 
called  higher  plane  curves. 

30.  A  transcendental  curve  may  in  general  be  intersected  by 
a  right  line  in  an  unlimited  number  of  points. 

Examples  of  the  more  common   and   useful  transcendental 


DEFINITIONS.  69 

curves  are  the  curves  of  the  trigonometric  functions,  as  the 
sine,  tangent ;  the  rolling  curves,  as  the  cycloid,  epicycloid, 
hypocycloid  ;  the  spirals,  as  the  involute  of  a  circle,  spiral  of 
Archimedes,  logarithmic  spiral. 

31.  Tangent.     A  tangent  to  a  curve  is  a  right  line  passing 
through  any  two  of  its  consecutive  points.     A  curve  is  always 
convex  towards  a  tangent  at  the  point  of  contact. 

32.  Normal.     A  normal  to  a  curve  is  a  right  line  drawn  _L 
to  a  tangent  at  the  point  of  tangency.     In  plane  curves  the 
normal  is  taken  in  the  plane  of  the  curve. 

33.  Axis  and  Vertex.     If  a  normal  divides  a  curve  into  two 
symmetrical  parts,  it  is  an  axis,  and  its  point  of  intersection 
with  the  curve  is  a  vertex. 

34.  Diameter.     If  we  draw  over  a  plane  curve  a  group  of  II 
chords  and  join  their  centres  successively  with  a  line,  the  latter 
is  a  straight  or  curved  diameter. 

35.  Osculating  Circle  and  Radius  of  Curvature.     The  circle 
which  passes  through  any  three  consecutive  points  of  a  curve 
is  an  osculating  circle,  and  its  radius  is  the  radius  of  curvature 
for  the  curve  at  the  middle  of  the  three  points.     The  radius  is 
usually  represented  by  p. 

36.  Space  Curves.     The  number  of  space  curves  is  unlim- 
ited, but  the  two  of   special   interest   are   the   helix   and   the 
spherical  epicycloid. 

37.  Helix.     The   Helix   is  generated   by  a  point  revolving 
about  a  fixed  right  line  called  its  axis.     The  generating  point 
remains  at  a  constant  distance  from  the  latter  and  has  a  velocity 
of  translation  in  its  direction  in  a  constant  ratio  to  its  angular 
velocity  about  it. 

38.  Spherical  Epicycloid.     The  spherical  epicycloid  is  gen- 
erated by  a  point  in  an  element  of  one  cone  of  revolution  roll- 
ing upon  another,   both  cones    having  the  same   vertex   and 
always  one  and  but  one  element  in  common. 


70  DESCRIPTIVE   GEOMETBY. 

SECTION  IV. 
Surfaces  in  General. 

39.  Surface.     A  surface  is  the  locus  of  the  different  posi- 
tions of  a  line,  called  the  generatrix,  or  it  is  the  envelope  of 
the   different   positions   of   other  surfaces.     The   law   for   the 
motion  must  state  whether   the  generating  figure  remains   of 
unchanging  form  or  not. 

40.  Directrix,  Directer.     Fixed  lines  along  which  the  gener- 
atrix glides,  and  fixed  surfaces  against  which  it  assumes  defin- 
ite positions,  are  called  directrices  and  directers. 

41.  Kinds  of  Surfaces.     Since   surfaces   in  general  involve 
the  idea  of  three  dimensions  in  space,  Analytic  Geometry  uses 
equations  involving  three  variables  to  designate  them  and  di- 
vides them  into  algebraical  and  transcendental  in  the  same  way 
that  it  divides  lines. 

42.  Algebraical  surfaces  are  divided  into  orders  and  classes! 
Nth  Order.     A  surface  of  the  nth  order  is  one  that  is  cut  by 

every  plane  in  a  line  of  the  nth   order.     The  surface   of  the 
first  order  is  the  plane. 

43.  Families   of  Surfaces.     Irrespective  of  these  divisions, 
surfaces  are  brought  together  in  families,  members  of  the  same 
family  having  something  in  common  in  their  generation. 

The  majority  of  surfaces  applied  in  the  arts  belong  to  two 
such  groups,  the  first  family  has  the  simplest  line,  —  the  right 
line,  —  for  generatrix ;  the  second  is  formed  by  the  simplest 
motion  of  any  line,  —  revolution  about  a  fixed  axis.  These 
groups  are  called  ruled  surfaces  and  surfaces  of  revolution. 
Several  special  forms  belong  to  both  families. 

44.  Ruled  Surface.     A  ruled  surface  is  the  locus  of  a  right 
line  moving  in  conformity  with  a  given  law. 

1  The  student  is  referred  to  Analytic  Geometry  of  three  dimensions,  for 
the  discussion  of  classes  of  surfaces. 


DEFINITIONS.  71 

There  are  three  subdivisions : 

1.  Plane. 

2.  Developable  surfaces,  or  such  as  have  every   two 

consecutive    positions  of   the  generatrix   in   the 
same  plane. 

3.  Warped  surfaces,  or  such  as  have  every  two  suc- 

cessive positions  of  the  generatrix  windschief. 
NOTE.     Any  two  right  lines  crossed  in  space,  but  not  intersecting,  are 
called  windschief. 

45.  Double  Curved   Surface.     Any  geometrical  surface  not 
belonging  to  the  ruled  surfaces  belongs  to  the  double  curved 
surfaces. 

SECTION  V. 
Developable  Surfaces. 

46.  Developable    Surface.     A  developable   surface   is   also 
the  envelope  of  a  plane  which  moves  in  accordance  with  some 
given  law,  but  the  simplest  generation  is  to  cause  a  right  line 
to  move  constantly  tangent  to  any  space  curve.    It  might,  there- 
fore, be  called  a  tangential  surface.     A  plane  element  of  this 
surface  will   be  the  portion  of  a  plane  included  between  two 
successive  and  intersecting  positions  of  the  generatrix.     There- 
fore the  surface  can  be  developed  into  a  plane  by  enlarging  the 
dihedral  angle  between  all  successive  plane  elements  to  180°. 
at  every  stage  keeping  the  plane  elements  already  developed 
fixed,  while  the  remaining  elements  are  consecutively  added 
thereto. 

47.  Developable   Surface  with  Helical  Directrix.     An  ex- 
ample of  the  developable  surface  is  that  with  a  helical  directrix, 
in  which  the  space  curve  along  which  the  generatrix  glides  is, 
as  the  name  indicates,  the  helix. 

48.  A   simple   developable   surface  arises  when    the   space 
curve  directing  the  generatrix  is  reduced  to  a  fixed  point.     In 
this  case  the  motion  of  the  generatrix  is  to  be  restricted  by 
another  condition. 


72  DESCRIPTIVE   GEOMETRY. 

49.  Cone.     A  cone  is  the  locus  of  a  right  line  which  always 
contains  a  fixed  point,  m,  while  it  passes  successively  through  all 
the  points  of  a  given  curved  line  or  directrix.     If  the  directrix 
is  a  plane  curve,  m  must  not  lie  in  the  plane  of  the  curve. 

When  the  directrix  is  a  curve  of  the  second  order,  the  cone 
is  a  surface  of  the  second  order. 

50.  Cylinder.     If  the  fixed  point  m  lies  in  infinity,  the  sides 
of   the  surface  are  parallel  and  the  cone  becomes  a  cylinder 
whose  order  is  determined  in  the  same  wa}7  as  that  of  the  cone. 

51.  Tangent  Plane.     A  tangent  plane  to  a  surface  at  a  given 
point  is  the  plane  which  contains  all  the  tangent  lines  to  the 
surface  at  that  point.     Any  two  of  these  lines  are  sufficient  to 
determine  the  plane.     In  general,  the  plane  tangent  to  a  devel- 
opable surface  at  a  given  point  upon  it,  is  most  readily  deter- 
mined by  the  element  through  the  point  of  tangency  and  the 
line  tangent  to  the  base  of  the  surface  at  the  point  in  which  the 
element  of  tangency  pierces  the  plane  of  the  base. 

52.  Shortest  Path.     The  shortest  path  upon  a  developable 
surface  between  two  points  in  that  surface,  is  the  right  line 
joining  those  points  when  the  surface  has  been  developed.     On 
the  cylinder  of  revolution  this  line  becomes  identical  with  the 
helix  and  the  loxodrome. 

SECTION  VI. 
Surfaces  of  Revolution. 

NOTE.  Before  taking  up  warped  surfaces,  we  will  consider  surfaces 
of  revolution,  as  one  warped  surface,  the  hyperboloid  of  one  sheet,  in  the 
form  most  applied  in  the  arts,  is  a  surface  of  revolution  and  can  be  most 
conveniently  treated  as  such. 

53.  Surface  of  Revolution.     A  surface  of  revolution  is  the 
locus  of  any  line  which  remains  unchanged  in  form  and  in  posi- 
tion with  reference  to  a  right  line  about  which  it  revolves  with- 
out a  motion  of  translation. 

54.  Axis.     The  fixed  line  is  called  the  axis  of  revolution, 
or  simply  the  axis. 


DEFINITIONS.  73 

55.  Parallels.     Every  plane  intersecting  the  surface  and  per- 
pendicular to  the  axis  cuts  out  a  circle.     All  such  circles  are 
called  parallels. 

56.  Meridian.     Every  plane  passed  through  the  axis  cuts 
out  a  curve  called  a  meridian.     All  meridians  of  the  same  sur- 
face of  revolution  are  equal  and  symmetrical  with  respect  to 
the  axis.     The  meridian  curve  of  a  surface  of  revolution  is  also 
called  its  profile. 

57.  Equator.     When  a  plane  of  symmetry  for   a  meridian 
curve  can  be  found  perpendicular  to  the  axis,  the  parallel  cut 
by  it  is  called  the  equator  of  the  surface. 

58.  Representation  of  Surfaces  of  Revolution.     By  the  defi- 
nition of  surface  of  revolution,  we  know  two  systems  of  lines, 
and  these  serve  to  represent  the  surface.     The  representation 
is  accomplished  most  simply  by  placing  the  axis  perpendicular 
to  H,  when  the  boundary  of  the  V  projection  will  be  a  meridian 
section,  while  the  H  projection  will  be  either  the  parallel  cut 
out  by  the  H  plane,  or  the  intersection  with  H  of  the  tangent 
horizontal  projecting  cylinder. 

59.  Circle  of  the  Gorge.     A  parallel  whose  radius  is  smaller 
than  that  of  any  other,  but  greater  than  zero,  is  a  circle  of  the 
gorge. 

60.  According  as  the  meridian  is  a  transcendental  or  alge- 
braical line   of   the   nth  order,  so  also  is  the  surface  of  the 
same  kind  and  order. 

61.  Surfaces  of  revolution  may  be  divided  into  orders,  as 
follows  :  — 

First  Order.    The  meridian  is  a  right  line  _L  to  the  axis  of 

revolution,  — plane. 
Second  Order.    The  meridian  is  composed  of : 

a)  Two  lines  parallel  to  and  equally  distant  from  the 

axis,  —  cylinder. 

b)  Two  lines  intersecting  the  axis  and  equally  inclined 

to  it,  —  cone. 

c)  A  circle  with  centre  in  the  axis,  —  sphere, 


74  DESCRIPTIVE   GEOMETRY. 

d)  An  ellipse  whose  minor  or  major  axis  lies  in  the 

axis    of     revolution,  —  the    oblate    or    prolate 
spheroid. 

e)  A   parabola  with   axis  as   axis   of    revolution, — 

paraboloid. 

f)  An  hyperbola  whose  real  axis  is  the  axis  of  revolu- 

tion, —  hyperboloid  of  double  sheet. 

g)  An  hyperbola  whose  imaginary  axis  is  the  axis  of 

revolution,  —  hyperboloid  of   single    sheet,     g) 
may  also  be  generated  by  revolving  one   right 
line  about  another  to  which  it  lies  windschief. 
Higher   Orders.    Other  kinds  of  revolution  of  lines  of  the 
second  order,  give  surfaces  of    the  fourth  order  often 
applied  in  the  arts  for  vases,  light-house  towers,   &c. 
The  conchoid  of   Nicomedes,  the  oval,  the  logarithmic 
curve,  the  cosine  curve   and  allied  forms,  the  cycloid, 
etc.,  generate  surfaces  of  revolution  much  used  in  the 
arts  and  which  are  algebraical  or  transcendental  accord- 
ing as  the  meridian  curves  are  the  one  or  the  other. 

62.  Tangent  Plane.     When  the  point  of  tangency  on  a  sur- 
face of    revolution  is    given,  the  tangent  plane    is   generally 
determined  most  readily  by  the  tangent  at  the  given  point  to 
the  parallel   and  the  tangent  through  the  same  point  to  the 
meridian. 

This  method  is  simplified  in  the  cone  and  cylinder,  while  in 
the  hyperboloid  of  revolution  of  one  sheet,  a  rectilinear  element 
passing  through  the  point  of  tangency  is  taken  from  each  of 
the  two  systems  of  generation. 

SECTION  VII. 
Warped  Surfaces. 

63.  Warped  Surface,     A  warped  surface  is  a  ruled  surface 
in  which  any  two  consecutive   rectilinear   elements  are  wind- 
schief.    Since  no  limit  can  be  placed  to  the  laws  which  shall 


DEFINITIONS.  75 

govern  the  motion  of  a  right  line  in  space,  none  can  be  placed 
to  the  number  of  possible  warped  surfaces.  In  general,  the 
generatrix  will  have  directing  points,  lines,  or  surfaces  which 
it  shall  intersect  or  intersect  at  a  given  angle ;  for  example, 
simply  touch  or  touch  so  as  to  cut  a  given  system  of  lines  at 
a  given  angle. 

64.  These  laws  can  be  variously  expressed  ;  as,  for  example, 
the  generatrix  shall  always  be  at  a  given  distance  from  a  fixed 
point,  means  also  that  it  shall  always  be  tangent  to  a  sphere  of 
given  radius. 

65.  The  simplest  law  for  the  generation  of  warped  surfaces 
is  that  the  generatrix  shall  glide  along  three  given  lines,  always 
intersecting  all  three.     If  every  two  consecutive  elements  inter- 
sect, the  surface  becomes  a  developable  one  ;  in  general  this  is 
not  so,  and  the  surface  is  warped. 

66.  Property  of  a  Plane  containing  an  Element  of  a  Warped 
Surface.     Since  each  element  of  a  warped  surface  is  in  general 
windschief  with  respect  to  every  other,  a  plane  containing  one 
element  will  be  pierced  by  every  other  element.     These  points 
of  piercing  together  form  a  curve  intersecting  the  given  ele- 
ment.    The  tangent  to  this  curve  at  the  point  where  it  inter- 
sects the  given  element,  together  with  the  element  itself,  forms 
two  intersecting  lines  tangent  to  the  surface  at  the  same  point. 
They  therefore  determine  the  plane  which  we  have  chosen,  with 
the  single  condition  that  it  shall  contain  an  element,  and  make 
of  it  a  tangent  plane.    Therefore,  in  general,  it  follows  that  any 
plane  containing  an  element  of  a  warped  surface  will  be  tan- 
gent to  the  surface  at  some  point  of  the  element. 

67.  Mutually  Tangent  Warped  Surfaces.     If   two  warped 
surfaces  have  an  element  in  common  and  are  tangent  to  each 
other  at  three  points,  a,  6,  c,  of  the  same,  then  they  are  tan- 
gent along  the  entire  element.    For  if  we  intersect  the  two  sur- 
faces at  the  given  points  by  three  planes,  the  latter  must  cut 
three  linear  directrices  from  each  surface.     Each  pair  of  direc- 
trices must  have  the  point  of    tangency  and  its  consecutive 


76  DESCRIPTIVE   GEOMETRY. 

point  in  common,  that  is,  aal9  bb^  c^ ;  therefore  these  pairs  of 
common  points  serve  to  determine  two  consecutive  positions  of 
the  generatrix.  It  follows  that  a  common  warped  surface  ele- 
ment will  be  determined,  and  that  a  plane  intersecting  the  sur- 
faces at  any  other  point  than  a,  6,  or  c,  of  the  common  element, 
must  cut  out  two  consecutive  points  common  to  each  surface. 

68.  Divisions  of  Warped  Surfaces.     A  linear  directrix  can 
be  a  right  line  or  a  curved  line  (the  latter  a  plane  or  space 
curve).     There  are,  therefore,  four  kinds  of  warped  surfaces, 
distinguished  as  follows  ;  the  directrices  are  : 

1)  Three  right  lines,  —  the  rryperboloid  of  one  sheet. 

2)  Two  right  lines  and  a  curved  line. 

3)  One  right  line  and  two  curved  lines. 

4)  Three  curved  lines. 

69.  Infinite  Directrices.     A  right  line  can  be  one  of  the  in- 
finite right  lines  of    space.     In  this  case  it  determines  with 
every  point  in  space  not  contained  in  itself  a  plane,  and  all 
the  planes  determined   by   it  and   finite  points   of  space   are 
parallel. 

If  two  infinite  right  lines,  or  two  parallel  right  lines,  are  used 
as  directrices,  they  will  have  an  infinite  point  in  common  and 
the  generatrix  must  alwa}*s  pass  through  this  point.  Therefore 
the  surfaces  formed  with  two  such  directrices  and  any  kind  of 
a  line  used  as  a  third  directrix,  will  be  planes  or  cones  (regard- 
ing the  cylinder  as  a  special  form  of  the  cone) .  If  the  two  in- 
finite directrices  were  curved  they  would  have  as  many  infinite 
points  of  intersection  as  the  orders  of  the  curves  would  allow  ; 
therefore,  with  a  third  directrix,  there  would  now  be  formed  a 
group  of  planes  or  cones. 

70.  But  One  Infinite  Directrix.     It  follows  that  a  warped 
surface  can  have  but  one  infinite  line  as  directrix,  and  since 
such  a  line  gives  rise  to  a  series  of    parallel  planes,  we  may 
take  any  one  of  these  as  a  plane  directer  and  every  element 
will  be  parallel  to  this  plane,  while  at  the  same  time  it  intersects 
the  other  two  directrices. 


DEFINITIONS.  77 

71.  Special    Group   of    Warped  Surfaces.      There  is  thus 
formed  a  special  group  of  warped  surfaces,  divided  as  follows  : 

The  generatrix  shall  be  always  parallel  to  a  given  plane  and 
shall  cut 

1)  Two  right  lines,  —  the  warped  plane,  or  hyperbolic 

paraboloid. 

2)  One  right  line,  one  curved  line,  —  the  warped  cone 

or  conoid. 

3)  Two  curved  lines,  —  the  warped  cylinder  or  cylin- 

droid. 

72.  Orders  of  Warped  Surfaces.     Among  warped  surfaces 
there  are  two  of  the  second  order,  the  hyperboloid  of  one  sheet 
formed  by  a  right  line  gliding  upon  three  right  lines,  no  one  of 
which  is  an  infinite  right  line  of  space  ;  the  hyperbolic  parabo- 
loid formed  by  a  right  line  gliding  upon  three  right  lines,  one 
of  which  is  an  infinite  right  line  of  space,  or  by  a  right  line 
always  parallel  to  a  given  plane  moving  along  two  other  wind- 
schief  right  lines. 

73.  Higher  Orders.    All  other  warped  surfaces  are  of  higher 
orders,    either    transcendental  or   algebraic,  according  to  the 
orders  and  kinds  of  the  directrices  and  generatrices. 

74.  Screw  Surface.     Among  the  most  important  of  those  of 
higher  order  is  the  screw  surface.     It  is  generated  by  a  right 
line    intersecting    at   a   constant   angle    and   revolving   about 
another  right  line,  the  point  of  intersection  having  a  velocity 
of    translation  along  the  linear  directrix  in   a  constant  ratio 
greater  than  zero  to  the  angular  velocity  of  rotation  of  the 
generatrix  about  it.     When  the  angle  of  "intersection  is  a  right 
angle,  the  screw  surface  thus  formed  is  also  a  warped  cone  or 
conoid,  applied  in  spiral  staircases,  etc. 


MA  THE  MA  TICS. 


Bowser's  Academic  Algebra.  A  complete  treatise  through  the  progressions,  includ- 
ing Permutations,  Combinations,  and  the  Binomial  Theorem.  Half  leather.  #1.25. 

Bowser's  College  Algebra.  A  complete  treatise  for  colleges  and  scientific  schools. 
Half  leather.  $1.65. 

Bowser's  Plane  and  Solid  Geometry.  Combines  the  excellences  of  Euclid  with 
those  of  the  best  modern  writers.  Half  leather.  $1.35. 

Bowser's  Plane  Geometry.    Half  leather.    85  cts. 

Bowser's  Elements  of  Plane  and  Spherical  Trigonometry.    A  brief  coursa 

prepared  especially  for  High  Schools  and  Academies.     Half  leather.    $1.00. 

Bowser's  Treatise  on  Plane  and  Spherical  Trigonometry.     An  advanced 

work  which  covers  the  entire  course  in  higher  institutions.     Half  leather.    $1.65. 

Hanus's  Geometry  in  the  Grammar  Schools.    An  essay,  together  with  illustrative 

class  exercises  and  an  outline  of  the  work  for  the  last  three  years  of  the  grammar  school. 
52  pages.     25  cts. 

Hopkin's  Plane  Geometry.     On  the  heuristic  plan.     Half  leather.     85  cts. 

Hunt's  Concrete  Geometry  for  Grammar  Schools.  The  definitions  and  ele- 
mentary concepts  are  to  be  taught  concretely,  by  much  measuring,  by  the  making  of 
models  and  diagrams  by  the  pupil,  as  suggested  by  the  text  or  by  his  own  invention, 
loo  pages.  Boards.  30  cts. 

Waldo's  Descriptive  Geometry.  A  large  number  of  problems  systematically  ar- 
ranged and  with  suggestions.  90  cts. 

The  New  Arithmetic.  By  300  teachers.  Little  theory  and  much  practice.  Also  aa 
excellent  review  book.  230  pages.  75  cts. 

For  Arithmetics  and  other  elementary  work  see  our  list  of  books  in  Number. 


D.   C.   HEATH    &   CO.,  PUBLISHERS, 

BOSTON.        NEW  YORK.        CHICAGO. 


ARITHMETIC. 

Aids  to  Dumber,  —  First  Series.     Teacher?  Edition. 

Oral  Work  —  One  to  ten.     25  cards  with   concise   directions.      By  ANNA    B. 
Principal  of  Training  School,  Lewiston,  Me.,  formerly  of  Rice  Training  School,  fio&lon. 
Retail  price,  40  cents. 

*Aids  to  Dumber.  —  First  Series.     Pupils*  Edition. 

Written  work.  —  One  to  ten.     Leatherette.     Introduction  price,  25  cents. 

Aids  to  Dumber.  —  Second  Series.     Teachers'  Edition. 

Oral  Work.  —  Ten  to  One  Hundred.     With  especial  reference   to   multiples  of  numbers 
from  i  to  10.     32  cards  with  concise  directions.     Retail  price,  40  cents. 

Aids  to  Cumbers.  —  Second  Series.     pupiii  Edition. 

Written  Work.  —  Ten  to  One  Hundred.     Leatherette.     Introduction  pricet  25  cents. 
The    Child's    t/Umber     Charts.       By  ANNA  B.  BADLAM. 


Manilla  card,  nx  14  inches.      Price,  5  cents  each  ;    $4.00  per  hundred. 

ChartS.     By  C.  P.  ROWLAND,  Principal  of  Tabor  Academy,  Marion,  Mass. 


For  rapid,  middle-grade  practice  work  on  the  Fundamental  Rules  of  Arithmetic.     Two 
cards,  8x9  inches.    Price,  3  cents  each  ;  or  $2.40  per  hundred. 


Number    CardS.      By  ELLA  M.  PIERCE,  of  Providence,  R.  I. 


For  Second  and  Third  Year  Pupils.     Cards,  7  x 9  inches.     Price,  3  cente  each;  or  $2.40 
per  hundred. 

ProHemS.      By  Miss  H.  A.  LUDDINGTON, 


Principal  of  Training  School,  Pawtucket,  R.  I.  ;  formerly  Teacher  of  Methods  and  Train. 
ing  Teacher  in  Primary  Department  of  State  Normal  School,  New  Britain,  Conn.. 
and  Training  Teacher  in  Cook  County  Normal  School,  Normal  Park,  111.  70  colored 
cards,  4x5  inches,  printed  OTI  both  sides,  arranged  in  9  sets,  6  to  10  cards  in  each  set. 
with  card  of  directions.  Retail  price,  65  cents. 


{Mathematical  Teaching  and  its  [Modern  [Methods. 

By  TRUMAN   HENRY  SAFFC 
Mass.     Paper.     47  pages.    1 

The  New  Arithmetic. 


By  TRUMAN   HENRY   SAFFORU,  Ph.  D.,   Professor  of  Astronomy,  Williams   College, 
Mass.     Paper.     47  pages.    Retail  price,  25  cents. 


By  300  authors.     Edited  by  SEYMOUR  EATON,  with  Preface  by  T.  H.  S  AFFORD,  Pro. 
fessor  of  Astronomy,  Williams  College,  Mass.     Introduction  price,  75  cents. 

D.    C.    HEATH    &    CO.,    Publishers, 

BOSTON,  NEW  YORK,  AND  CHICAGO. 


DRAWING  AND  MANUAL  TRAINING. 


Johnson's  Progressive  Lessons  in  Needlework.     Explains  needlework  from  its 

rudiments  and  gives  with  illustrations  full  directions  for  work   during  six  grades.     117 
pages.     Square  8vo.     Cloth,  $1.00.     Boards,  60  cts. 

Seidel's  Industrial  Instruction  (Smith).  A  refutation  of  all  objections  raised  against 
industrial  instruction.  1 70  pages.  90  cts. 

Thompson's  Educational  and  Industrial  Drawing. 

Primary  Free-Hand  Series  (Nos.  1-4).     Each  No.,  per  doz.,$i.oo. 
Primary  Free-Hand  Manual.     114  pages.     Paper.     40  cts. 
Advanced  Free-Hand  Series  (Nos.  5-8).     Each  No.,  per  doz.,  $1.50. 
Model  and  Object  Series  (Nos.  1-3).     Each  No.,  per  doz.,  $1.75. 
Model  and  Object  Manual.     84  pages.     Paper.     35  cts. 
./Esthetic  Series  (Nos.  1-6).     Each  No.,  per  doz.,  $1.50. 
./Esthetic  Manual.     174  pages.     Paper.     60  cts. 
Mechanical  Series  (Nos.  1-6).     Each  No.,  per  doz.,  $2.00. 
Mechanical  Manual.     172  pages.     Paper.     75  cts. 
Models  to  accompany  Thompson's  Drawing: 

Set  No.  I.     For  Primary  Books,  per  set,  40  cts. 

Set  No.  II.     For  Model  and  Object  Book  No.  i,  per  set,  oo  cts. 

Set  No.  III.     For  Model  and  Object  Book  No.  2,  per  set,  50  cts. 

Thompson's  Manual  Training,  NO.  I.  Treats  of  Clay  Modelling,  Stick  and 
Tablet  Laying,  Paper  Folding  and  Cutting,  Color,  and  Construction  of  Geometrical 
Solids.  Illustrated.  66  pages.  Large  8vo.  Paper.  30  cts. 

Thompson's  Manual  Training,  NO.  2.  Treats  of  Mechanical  Drawing,  Clay- 
Modelling  in  Relief,  Color,  Wood  Carving,  Paper  Cutting  and  Pasting.  Illustrated. 
70  pp.  Large  8vo.  Paper.  30  cts. 

Waldo's  Descriptive  Geometry.  A  large  number  of  problems  systematically  ar- 
ranged, with  suggestions.  85  pages.  90  cts. 

Whitaker's  HOW  tO  Use  WOOd  Working  Tools.  Lessons  in  the  uses  of  the 
universal  tools:  the  hammer,  knife,  plane,  rule,  chalk-line,  square,  gauge,  chisel,  saw, 
and  auger.  104  pages.  60  cts. 

Woodward's  Manual  Training  School.  Its  aims,  methods,  and  results;  with" 
detailed  courses  of  instruction  in  shop-work.  Fully  illustrated.  374  pages.  Octavo.  $2.00. 

Woodward's  Educational  Value  of  Manual  Training.    Sets  forth  more  clearly 

and  fully  than  has  ever  been  done  before  the  true  character  and  functions  of  manual  train- 
ing  in  education.     96  pages.     Paper.     25  cts. 

Sent  postpaid  by  mail  on  receipt  of  price. 


D.    C.    HEATH    &    CO.,    PUBLISHERS, 

BOSTON.        NEW  YORK.        CHICAGO. 


SCIENCE. 

Shaler'S    First  Book  in   Geology.      For  high  school,  or  highest  class  in  grammar 
school,     jii.io.     Bound  in  boards  for  supplementary  reader.    70  cts. 

World    Of   Matter.       A  Guide  to  Mineralogy  and  Chemistry.     $x.oo. 


Inorganic   Chemistry.      Descriptive  and  Qualitative;  experimental  and 
inductive;  leads  the  student  to  observe  and  think.    For  high  schools  and  colleges.    $1.25. 

Shepard's  Briefer  Course  in  Chemistry  ;  with  Chapter  on  Organic 

Chemistry.     Designed  for  schools  giving  a  half  year  or  less  to  the  subject,  and  schools 
limited  in  laboratory  facilities.     90  cts. 

Shepard's   Organic    Chemistry.      The  portion  on  organic  chemistry  in  Shepard's 
Briefer  Course  is  bound  in  paper  separately.     Paper.    30  cts. 


Laboratory   Note-Book.      Blanks  for  experiments:  tables  for  the  re. 
actions  of  metallic  salts.     Can  be  used  with  any  chemistry.     Boards.     40  cts. 

Benton's  Guide  to  General  Chemistry.    A  manual  for  the  laboratory.  4octs. 


Organic    Chemistry.       ^n  Introduction  to  the  Study  of  the  Compounds 
of  Carbon.     For  students  of  the  pure  science,  or  its  application  to  arts.    $1.30. 

Orndorff's   Laboratory    Manual.      Containing  directions  for  a  course  of  experiments 
in  Organic  Chemistry,  arranged  to  accompany  Remsen's  Chemistry.     Boards.     40  cts. 

Colt's   Chemical   Arithmetic.      With   a  short  system  of  Elementary  Qualitative 
Analysis      For  high  schools  and  colleges.     60  cts. 

Grabfield  and  Burns'  Chemical  Problems.    For  preparatory  schools.  60  cts. 

€hllte'S   Practical   PhysiCS.      A  laboratory  book  for  high  schools  and  colleges  study- 
ing ynysics  experimentally.     Gives  free  details  for  laboratory  work.    $1.25. 


Practical    Zoology.      Gives  a  clear  idea  of  the  subject  as  a  whole,  by  the 
careful  study  of  a  few  typical  animals.    90  cts. 

Boyer's  Laboratory  Manual  in  Elementary  Biology.    A  guide  to  the 

study  of  animals  and  plants,  and  is  so  constructed  as  to  be  of  no  help  to  the  pupil  unless 
he  actually  studies  the  specimens. 

Clark's  Methods  in  MicrOSCOpy.  This  book  gives  in  detail  descriptions  of  methods 
that  will  lead  any  careful  worker  to  successful  results  in  microscopic  manipulation.  $1.60. 

Spalding'3  Introduction  tO  Botany.  Practical  Exercises  in  the  Study  of  Plants 
by  the  laboratory  method.  90  cts. 

Whiting's  Physical  Measurement.  Intended  for  students  in  Civil,  Mechani- 
cal and  Electrical  Engineering,  Surveying,  Astronomical  Work,  Chemical  Analysis,  Phys- 
ical Investigation,  and  other  branches  in  which  accurate  measurements  are  required. 

I.     Fifty  measurements  in  Density,  Heat,  Light,  and  Sound.     $1.30. 
II.     Fifty  measurements  in  Sound,  Dynamics,  Magnetism,   Electricity.    #1.30. 
III.     Principles  and  Methods  of  Physical  Measurement,  Physical  Laws  and  Princi- 

ples, and  Mathematical  and  Physical  Tables.     $1.30. 

IV.  Appendix  for  the  use  of  Teachers,  including  examples  of  observation  and  re- 
duction. Part  IV  is  needed  by  students  only  when  working  without  a  teacher. 
$1-30. 

Parts  I-III,  in  one  vol.,  $3.25.     Parts  I-IV,  in  one  vol.,  $4.00. 

Williams'  S  Modern  Petrography.  An  account  of  the  application  of  the  micro- 
scope to  the  study  of  geology.  Paper.  25  cts. 

For  elementary  works  see  our  list  of  looks  in  Elementary  Science. 

D.    C.   HEATH   &   CO.,    PUBLISHERS. 

BOSTON.        NEW  YORK.        CHICAGO. 


ELEMENTARY  SCIENCE. 


Bailey's  Grammar  School  PhysiCS.  A  series  of  inductive  lessons  in  the  elements 
of  the  science.  In  press. 

Ballard'S  The  World  Of  Matter.  A  guide  to  the  study  of  chemistry  and  mineralogy; 
adapted  to  Uie  general  reader,  for  use  as  a  text-book  or  as  a  guide  to  the  teacher  in  giving 
object-lessons.  264  pages.  Illustrated.  |>i.oo. 

Clark's  Practical  Methods  in  MicrOSCOpy.  Gives  in  detail  descriptions  of  methods 
that  will  lead  the  careful  worker  to  successful  results.  233  pages.  Illustrated.  $1.60. 

Clarke's  Astronomical  Lantern.  Intended  to  familiarize  students  with  the  constella- 
tions by  comparing  them  with  fac-similes  on  the  lantern  face.  With  seventeen  slides, 
giving  twenty-two  constellations.  $4.50. 

Clarke's  HOW  tO  find  the  Stars.  Accompanies  the  above  and  helps  to  an  acquaintance 
with  the  constellations.  47  pages.  Paper.  15  cts. 

Guides  for  Science  Teaching.  Teachers'  aids  in  the  instruction  of  Natural  History 
classes  in  the  lower  grades. 

I.     Hyatt's  About  Pebbles.     26  pages.     Paper.     10  cts. 
II.     Goodale's  A  Few  Common  Plants.     61  pages.     Paper.     20  cts. 

III.  Hyatt's  Commercial  and  other  Sponges.    Illustrated.    43  pages.  Paper.   20  cts. 

IV.  Agassiz's  First  Lessons  in  Natural  History.     Illustrated.     64  pages.     Paper. 

25  cts. 

V.     Hyatt's  Corals  and  Echinoderms.     Illustrated.     32  pages.    Paper.     30  cts. 
VI.     Hyatt's  Mollusca.     Illustrated.     65  pages.     Paper.     30  cts. 
VII      H'-att's  Worms  and  Crustacea.     Illustrated.     68  pages.     Paper.     30  cts. 
VIII      B /att's  Insecta.     Illustrated.     324  pages.     Cloth.     $1.25. 
XII-     <  rosby's  Common  Minerals  and  Rocks.     Illustrated.    200  pages.     Paper,  40 
cts.     Cloth,  60  cts. 

XIII  <<ichard's  First  Lessons  in  Minerals.     50  pages.     Paper.     10  cts. 

XIV  Bowditch's  Physiology.     58  pages.     Paper.     20  cts. 

XV      Clapp's  36  Observation  Lessons  in  Minerals.    80  pages.     Paper.     30  cts. 
XVI      Phenix's  Lessons  in  Chemistry.     1 n  press. 
Pupils   Note-Book  to  accompany  No.  15.     10  cts. 

Rice's  Science  Teaching  in  the  School.  With  a  course  of  instruction  in  science 
for  the  lower  grades.  46  pag  s.  Paper.  25  cts. 

Ricks*?  Natural  History  Object  Lessons.  Supplies  information  on  plants  and 
their  products,  on  animals  and  their  uses,  and  gives  specimen  lessons.  Fully  illustrated. 
332  pages.  $1.50. 

Ricks's  Object  Lessons  and  How  to  Give  them. 

Volume  I.     Gives  lessons  for  primary  grades.     200  pages.     90  cts. 

Volume  II.  Gives  lessons  tor  grammar  and  intermediate  grades.  212  pages.     90  cts. 

Shaler's  First  Book  in  Geology.  For  high  school,  or  highest  class  in  grammar  school. 
^72  pages.  Illustrated.  $1.00. 

Shaler's  Teacher's  Methods  in  Geology.  AD  aid  to  the  teacher  of  Geology. 
74  pages.  Paper.  25  cts. 

Smith's  Studies  in  Nature.  A  combination  of  ratural  history  lessons  and  language 
work.  48  pages.  Paper.  15  cts. 

Sent  by  mail  postpaid  on  receipt  of  price.     See  also  our  list  of  books  in  Science. 


D.    C.    HEATH   &    CO.,    PUBLISHERS, 

BOSTON.        NEW  YORK.        CHICAGO. 


EDUCATION. 


History  of  Pedagogy.      "  The  best  and  most  comprehensive  history  of 
Education  in  English."  —  Dr.  G.  S.  HALL.     $1.75. 
Compayre"'s  Lectures  On  Teaching.      "The  best  book  in  existence  on  thr  theory  and 
practice  of  education."  —  Supt.  MACALISTER,  Philadelphia.     $1.75. 

Compayre"'s  Psychology  Applied  tO  Education.  A  clear  and  concise  statement 
of  doctrine  ai;d  application  on  the  science  and  art  of  teaching,  go  cts. 

De  Garmo's  Essentials  Of  Method.  A  practical  exposition  of  methods  with  illustra- 
tive outlines  of  common  school  studies.  65  cts. 

De  Garmo's  Lindner's  Psychology.  The  best  Manual  ever  prepared  from  the 
Herbartian  standpoint.  $1.00. 

Gill'S  Systems  Of  Education.  "  It  treats  ably  of  the  Lancaster  and  Bell  movement 
in  education,  —  a  -very  important  phase."  —  Dr.  W.  T.  HARRIS.  $1.25. 

Hall's  Bibliography  of  Pedagogical  Literature.     Covers  every  department  of 

education.     Interleaved,  *$2.oo.     $1.50. 

Herford'S  Student's  Froebel.  The  purpose  of  this  little  book  is  to  give  young  people 
preparing  to  teach  a  brief  yet  full  account  of  Froebal's  Theory  of  Education.  75  cts. 

Malleson's  Early  Training  of  Children.     "The  best  book  for  mothers  I  ever 

read."  —  ELIZABETH  P.  PEABODY.     75  cts. 

Marwedel'S    ConsciOUS   Motherhood.     The  unfolding  of  the   child's   mind  in   the 

cradle,  nursery  and  Kindergarten.     $2.00. 
Newsholme's  School  Hygiene.     Already  in  use  in  the  leading  training  colleges  in 

England.     75  cts. 

Peabody's  Home,  Kindergarten,  and  Primary  School.    "The  best  booi 

side  of  the  Bible  that  I  ever  read."  — A  LEADING  TEACHER.     $1.00. 

Pestalozzi's  Leonard  and  Gertrude.    "  if  we  except  '  Emile '  only,  no  mm 

portant   edurationnl  book  has  appeared  for  a  century  and  a  half  than  '  Leonard  an- 
trude.'  "  —  The  Nation.     90  cts. 

RadestOCk'8  Habit  in  Education.  "  It  will  prove  a  rare  '  find'  to  teachers  v  - 
seeking  to  ground  themselves  in  the  philosophy  of  their  art."  —  E.  H.  RUSSELL,  V 
ter  Normal  School.  75  cts. 

Richter's  Levana  ;  or,  The  Doctrine  of  Education.     "A  spirited  and  sc 

book."  — Prof.  W.  H.  PAYNE.    $1.40. 

ROSmini'S   Method  in  Education.     "The  most  important   pedagogical  work 
written."  —  THOMAS  DAVIDSON.    $1.50. 

Rousseau's  Emile.      "  Perhaps  the  most  influential  book  ever  written  on  the  sut 

Education."  —  R.  H.  QUICK.     90  cts. 
Methods  Of  Teaching  Modern  Languages.      Papers  on  the  value  and  on  m         .is 

of  teaching  German  and  French,  by  prominent  instructors.     90  cts. 

Sanford's  Laboratory  Course  in  Physiological  Psychology.     The       rse 

includes  experiments  upon  the  Dermal  Senses,  Static  and  Kinaesthetic  Senses,      aste, 
Smell,  Hearing,  Vision,  Psychophysic.     In  Press. 

Lange's  Apperception :  A  monograph  on  Psychology  and  Pedagogy.    .;rans- 

lated  by  the  members  of  the  Herbart  Club,  under  the  direction  of   President  Charles 
DeGarmo,  of  Swarthmore  College.     $1.00. 

Herbart's  Science  Of  Education.  Translated  by  Mr.  and  Mrs.  Felken  with  a  pref- 
ace by  Oscar  Browning.  $1.00. 

Tracy's   Psychology   Of  Childhood.     This  is  the  first  £*«<fr<z/  treatise  covering  in  a 
scientific  manner  the  whole  field  of  child  psychology.     Octavo.     Paper.     75  cts. 
Sent  by  mail,  postpaid,  on  receipt  of  price. 

D.    C.    HEATH    &    CO.,    PUBLISHERS, 

BOSTON.        NEW  YORK.        CHICAGO. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


JUL     221S46 

REC'D  L^ 

EEJB  2  -  log] 

NOV  11  vnifi 

'.     •-.'       rcTvl  ! 

.,•  -                         j 

:V49T 

2lOct'49Al 

mMqr'SOCS 

lUiVldi  ^^ 

,     -nM":  •   : 

REC'D  LD 

NOV   71957 

g-MAB'eiAO 

LD  21-100m-7,'40  (6936s) 

THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


